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COMPRESSED AIR 



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COMPRESSED AIR 



THEORY AND COMPUTATIONS 



BY 



ELMO G. HARRIS, C.E. 

PROFESSOR OF CIVIL ENGINEERING, MISSOURI SCHOOL OF MINES, 
IN CHARGE OF COMPRESSED AIR AND HYDRAULICS; 
MEMBER OF AMERICAN SOCIETY OF 
CIVIL ENGINEERS 



Second Edition 
Revised and Enlarged 



McGRAW-HILL BOOK COMPANY, Inc. 

239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Ltd. 
6 & 8 BOUVERIE ST., E. C. 

1917 



\c\vn 



Copyright, 1910, and 1917, by the 
McGraw-Hill Book Company, Inc. 



MAY 28 1917 

THE MAPLE PRESS YORK PA 



I 




**T3.A467199 



PREFACE TO SECOND EDITION 

After five years trying-out of the first edition the second has 
been prepared with the view to eliminate all errors and ambigui- 
ties and to add matter of value where possible without burdening 
the text with illustrations and descriptions of matter of only tem- 
porary value, such as machines and devices that are in use today 
but may be succeeded by better ones in a few years. It is the 
author's opinion that current practice, in the general form of 
machines and their details, can best be studied in trade circulars, 
of which there are many very creditable productions illustrating 
and describing a greater variety of machines than can possibly 
be shown in a text-book. 

A new chapter has been added on centrifugal fans and turbine 
compressors. The author has found a need for a clear, concise 
presentation of the theory underlying such machines, and be- 
lieves that a more general knowledge of the technicalities of the 
subject will lead to material betterment of the cheaper forms of 
fans that make up the greater portion of the total in use. 

Appendix B, on design of Logarithmic charts, should be welcome 
to most students since such matter has not appeared in text- 
books in common use. 

Compressed air has long held the field for rock drilling under- 
ground, though electricity has several times attempted to get into 
that business. At one time it seemed that compressed air would 
prove the best motive power for underground pumps, but in more 
recent years the improvements in centrifugal pumps seem to give 
electricity the advantage. 

In general, it may be assumed that where rotation is desired 
electricity will have the advantage, while where rapid reciprocat- 
ing motion is desired air will have the advantage. In the latter 
class are all kinds of pneumatic hammers, which have revolution- 
ized several industries since they have been introduced. 

It is not the intention to enumerate here the applications of 
compressed air. It is a very versatile, willing and good-natured 
servant. It offers a fascinating field for the inventor and its 
usefulness and already numerous applications will surely increase. 

Rolla, Mo., E. G. Harris. 

April, 1917. 



PREFACE TO FIRST EDITION 

This volume is designed to present the mathematical treat- 
ment of the problems in the production and application of com- 
pressed air. 

It is the author's opinion that prerequisite to a successful 
study of compressed air is a thorough training in mathematics, 
including calculus, and the mathematical sciences, such as 
physics, mechanics, hydraulics and thermodynamics. 

Therefore no attempt has been made to adapt this volume to 
the use of the self-made mechanic except in the insertion of more 
complete tables than usually accompany such work. Many 
phases of the subject are elusive and difficult to see clearly even 
by the thoroughly trained ; and serious blunders are liable to occur 
when an installation is designed by one not well versed in the 
technicalities of the subject. 

As one advocating the increased application of compressed air 
and the more efficient use where at present applied, the author has 
prepared this volume for college-bred men, believing that such 
only, and only the best of such, should be entrusted with the 
designing of compressed-air installations. 

The author claims originality in the matter in, and the use of, 
Tables I, II, III, V, VI, VII and IX, in the chapter on friction 
in air pipes and in the chapter on the air-lift pump. 

Special effort has been made to give examples of a practical 
nature illustrating some important points in the use of air or 
bringing out some principles or facts not usually appreciated. 

Acknowledgment is herewith made to Mr. E. P. Seaver for 
tables of Common Logarithms of Numbers taken from his 
Handbook. 



CONTENTS 

Page 

Peeface V 

Symbols . xi 

Formulas xiii 

Introduction xv 

CHAPTER I. 

Art. 1. Formulas for Work. — Temperature Constant .... 1 

Art. 2. Formula for Work. — Temperature Varying 3 

Art. 3. Formula for Work. — Incomplete Expansion. .... 7 

Art. 3a. Work as Shown by Indicator Diagrams 8 

Art. 4. Effect of Clearance. — In Compression 9 

Art. 5. Effect of Clearance and Compression in Expansion 

Engines 14 

Art. 6. Effect of Heating Air as it Enters Cylinders 17 

Art. 7. Change of Temperature in Compression or Expansion 18 

Art. 8. Density at Given Temperature and Pressure .... 19 

Art. 8a. Weight of Moist Air 19 

Art. 9. Cooling Water Required . . i 20 

Art. 10. Reheating and Cooling 21 

Art. 11. Compounding 23 

Art. 12. Proportions for Compounding 25 

Art. 13. Work in Compound Compression 27 

Art. 14. Work under Variable Intake Pressure . 27 

Art. 15. Exhaust Pumps . . • 29 

Art. 16. Efficiency when Air is Used without Expansion ... 30 

Art. 17. Variation of Free Air Pressure with Altitude .... 31 

CHAPTER II. Measurement op Air. 

Art. 18. General Discussion 33 

Art. 19. Apparatus for Measuring Air 34 

Art. 20. Measurement by Standard Orifice 35 

Art. 21. Formula — Standard Orifice under Standard Conditions 35 

Art. 22. Apparatus for Measuring Air at Atmospheric Pressure 37 

Art. 23. Coefficients for Large Orifices 38 

Art. 24. Apparatus for Measuring Air under Pressure with 

Standard Orifice 41 

Art. 25. Coefficients and Orifice Diameters for Measurements 

at High Pressure 43 

Art. 26; Discharge of Air through Orifices — Considerable Drop 

in Pressure 45 

Art. 27. Air Measurement in Tanks . 46 

CHAPTER III. Friction in Air Pipes. 

Art. 28. General Discussion 49 

Art. 29. The Formula for Practice 49 

ix 



x CONTENTS 

Page 

Art. 30., Theoretically Correct Friction Formula 57 

Art. 31. Efficiency of Power Transmission by Compressed Air. 60 

CHAPTER IV. Other Air Compressors. 

Art. 32. Hydraulic Air Compressors — Displacement Type . . 63 

Art. 33. Hydraulic Air Compressors — Entanglement Type . . 64 

Art. 34. Centrifugal or Turbo-air Compressors 66 

CHAPTER V. Special Applications of Compressed Air. 

Art. 35. Return-air System 68 

Art. 36. Return-air Pumping System 69 

Art. 37. Simple Displacement Pump 75 

CHAPTER VI. The Air-lift Pump. 

Art. 38. General Discussion 76 

Art. 39. Theory of the Air-lift Pump 76 

Art. 40. Design of Air-lift Pumps 78 

Art. 41. The Air-lift as a Dredge Pump 83 

Art. 42. Testing Wells with the Air-lift 84 

Art. 43. Data on Operating Air-lifts 85 

CHAPTER VII. Receivers and Storage of Compressed Air. 

Art. 44. Receivers for Suppressing Pulsations Only 87 

Art. 45. Receivers — Some Storage Capacity Necessary. ... 88 

Art. 46. Hydrostatic — Compressed Air Reservoirs 89 

CHAPTER VIII. Fans. 

Art. 47. Introductory 91 

Art. 48. Purely Centrifugal Effects 93 

Art. 49. Impulsive or Dynamic Effects .......... 95 

Art. 50. Discharging Against Back Pressure 96 

Art. 51. Designing 98 

Art. 52. Testing 100 

Art. 53. Suggestions 103 

CHAPTER IX. Centrifugal or Turbo-air Compressors. 

Art. 54. Centrifugal Compression of an Elastic Fluid .... 105 

Art. 55. Effect" of Picking up the Fluid . 106 

Art. 56. Working Formula 107 

Art. 57. Suggestions 110 

Art. 58. Proportioning Ill 

CHAPTER X. Rotary Blowers. 

Art. 59. General 113 

CHAPTER XL Examples and Exercises. 

Art. 60. Introductory 115 

Tables • . . . . 127 

Appendix A. Drill Capacity Tables 169 

Appendix B. Design of Logarithmic Charts 172 

Appendix C. Determination of Friction Factors 179 

Appendix D. Oil Differential Gage 187 

Index 191 



SYMBOLS 

For ready reference most of the symbols used in the text are assembled 
and defined here. 

p = intensity of pressure (absolute), usually in pounds per square foot. 
Compressed-air formulas are much simplified .by using pressures 
and temperatures measured from the absolute zero. Hence 
where ordinary gage pressures are given, p = gage pressure + 
atmospheric pressure. In the majority of formulas p must be 
in pounds per square foot, while gage pressures are given in 
pounds per square inch. Then p = (gage pressure + atmos- 
pheric pressure in pounds per square inch) X 144. 
v = volume — usually in cubic feet. 

Where sub-a is used, thus p a , v a , the symbol refers to free air 
conditions. 

, . . higher pressure 

r = ratio ot compression or expansion = ^- — 

lower pressure 

The lower pressure is not necessarily that of the atmosphere. 

t = absolute temperature = Temp. F. + 460.6. 

n = an empirical exponent varying from 1 to 1.41. 

log e = hyperbolic logarithm = (common log.) X 2.306. 

W = work — usually in foot-pounds. 

Q = weight of air passed in unit time. 

w = weight of a cubic unit of air. 

Other symbols are explained where used. 



INDEX TO FORMULAS 

Number Formula Art. Page 

1. W = pivi log, r; isothermal compression 1 2 

2. W = 63.871 logio r for one pound at 60° 1 2 

3. — ^ = -^ ; physical law of gases 2 3 

P2V2 t 2 

4. pv = 53.35 t for one pound 2 3 

5. pii>i" = pxVz™ = pzV2 n ; observed law of gases 2 3 

„ w P2W2 — PlVi . n 

6. W = 7 h P2^2 — Pa^o 2 4 

n — I 

7. TF = -^-- (p 2 y 2 - PaVa ) 2 4 

w — 1 

7a. Tf = - 53.35 (i 2 — h) for one pound 2 4 

n - 1 

8. IF = —^ p„« (r w - l) 2 5 

n - 1 

8d. TF = T (^J-j) 53.35 (r B - l) 1 < = kt, for one pound 2 6 

9. TP" = _ 1 1 + p2?>2 — p a vi ; incomplete expansion ... 3 8 

10. m.e.p. = 2.3 p a logio r 4a 12 

w - 1 

10a. m.e.p. = — r p a ( r M — 1 1 4a 12 

10b. m.e.p. = p 2 I — _ 1 ) — p a 4a 12 

( n) 

11. E = 1 + c\l — r / ; volumetric efficiency 4b 13 

/vi\ n ~ 1 "— ^ 

12. i 2 = h .Ml - - iir » 7 18 

7} 

13. w = go og . ; dry air 8 19 

00.00 t 

13c. w> = —— — (m — 0.38 Hq) ; moist air 8a 20 

14. 0*2 = —^i- and d 3 = — ; compounding 12 26 



n-\ 



15. W = — ^-r (p a Va) In ? — l) X 2 ; compound two-stage 13 27 

n- 1 

16. PT = — ^7 p a Wo m — 1] X 3 ; compound three-stage 13 27 



xni 



36 



XIV INDEX TO FORMULAS 

Number Formula Art. Page 

17. m = . tI^ — r — g , • exhaust pump 15 30 

log (V + v) — log V 

r — 1 

18. E = — -. ; efficiency without expansion 16 31 

r log e r 

19. p a = .4912[w - 0.0001 (F - 32)] ; pressure by barometer. 17 32 

20. logio p a = 1.4687 — -199 a/™ , Anr,\ ! pressure at elevation 17 32 

21. Q = c 0.1639 d 2 \l- rp a ) orifice measure 21 

21a. w a = 1.321 -; weight by mercury column 21 37 

22. di = -y ; orifice under pressure 25 44 

I v a - 

26. / = c -== — : friction in pipes 29 50 

a 5 r 

I ch)a 2 \ y$ 

27. d = I - ° ) ; friction in pipes 29 51 

O0 , 0.102 5 Zt>.«. f ..... 00 K1 

28. / = ir— : — : friction in pipes 29 51 

I v a - 
30. logio Pi = logio pi — c 2 -« -j — ; friction in pipes 30 58 

32. E = . 2 ; efficiency of transmission 31 60 

logri' 

loff r 

33. E = — ^-r-; efficiency-hydraulic air compression. 32 63 

o, v a ■ w(h + hi) . .... .„ on 

34. ~ = ^r — t — — J air lift pump 40 80 

Q E p a \og e r 

36. gf = Jl\ + h ; air lift pump 40 80 

Q 35 logio? 1 ' 

37. H = ; fans — centrifugal pumps 49 96 

„_ , „ w a lu~ + uV cos 0\ ,-"■. . CE in „ 

37a. loff e i?i = — ( ) turbo-air compression .... 55 107 

Pa\ g 1 

38. R n = Ri n and log R n = n log Ei ; turbo-air compression . . 55 107 

39. logx„ Ri = ( 2 . 3 x 5 3,3 5 x t x g ) ^ = ***i turbo-air com- 

pression 56 108 



INTRODUCTION 

Compressed Air is a manufactured product of considerably 
greater value than the things used and consumed in its manu- 
facture. The things used in its production are power, machinery, 
and superintendence — chiefly power. Since the compressed air 
is then more costly than the power applied in its production it is 
but reasonable that we should give as much attention to its 
efficient use, or more, than we would to the use of steam, water 
power or electric energy. Yet in much of the practice in using 
compressed air this anticipation does not seem to be realized. 
This may be due to the fact that the exceeding convenience and 
safety of compressed air, and its labor-saving qualities in many 
applications, make efficiency as measured by foot-pounds, a 
secondary consideration; and from this a habit of wastefulness is 
formed. A further explanation may be found in its harmlessness 
and general good nature. A leaky air pipe, or excessive use at a 
motor, does not scald, suffocate, nor becloud the view as with 
steam; nor shock, burn nor start a fire as with electricity, nor 
flood and foul the premises as with water. Hence the user is apt 
to tolerate wastes of compressed air that should be checked to 
save the coal pile. 

In some lines of industry compressed air is supreme, in others 
electricity, and still in others steam, and water, each being specially 
adapted to do certain things better than any other, but in some 
lines the winner has not been decided, or even though decided in 
so far as present methods apply there may, any day, arise fresh 
competition by the invention of new devices or processes. 



COMPRESSED AIR 



CHAPTER I 
FORMULAS FOR WORK 

Art. 1. Temperature Constant or Isothermal Conditions. — 

From the laws of physics (Boyle's Law) we know that while 
the temperature remains unchanged the product pv remains 
constant for a fixed amount (weight) of air. Hence to determine 
the work done on, or by, air confined in a cylinder, when condi- 
tions are changed from piVi to p%v 2 we can write 

PlVl = p x v x = P2V2, 
sub x indicating variable intermediate conditions. 




5 



TF 



rfr 



3 



Fig. 1. 



Whence p x = and dW = p x Adl — p x dv x since Adl = dv; 

Vx 

dv 
A being the area of cylinder, therefore dW = piVi — -, and 

v x 

work of compression or expansion between points B and C (Fig. 
1) is the integral of this, or 

1 



2 COMPRESSED AIR 

C V1 dv x 

W = PiVi I — - = P1V1 (l0g e Vi - log e V 2 ) 

= piVi log e -^ = pivi log e -* = pivi log e r = p 2 v 2 log e r. 
v 2 Pi 

Note that this analysis is only for the work against the front 
of the piston while passing from B to C. To get the work done 
during the entire stroke of piston from B to D we must note that 
throughout the stroke (in case of ordinary compression) air is 
entering behind the piston and following it up with pressure p\. 
Note also that after the piston reaches C (at which time valve / 
opens) the pressure in front is constant and = p 2 for the remainder 
of the stroke. Hence the complete expression for work done by, 
or against, the piston is 

PiVi log e r - piVi + p 2 v 2 ; 

but since piVi = p 2 v 2 , the whole work done is 

W = piVi log e r or p 2 v 2 log e r (1) 

Note that the operation may be reversed and the work be done 
by the air against the piston, as in a compressed-air engine, with- 
out in any way affecting the formula. 

Forestalling Art. 2, Eq. (4), we may substitute for pv in Eq. (1) 
its equivalent, 53.35 1, for 1 lb. of air and get for 1 lb. of dry air 

W = 53.35 1 Xlog e r (la) 

This may be adopted for common logs by multiplying by 2.3026. 
It then becomes 

W = (122.83 logio r) t (16) 

log 122.83 = 2.08930. 

Note that in solving by logs the log of log r must be taken. 
Values of the parenthesis in Eq. (16) are given in Table I, 
column 11. 

For the special temperature of 60°F. (16) becomes for 1 lb. of 
air 

W = 63,871 logio r (2) 

log 63,871 = 4.80536. 

Note that for moist air the coefficient in (la) is greater, being 
53.87 for saturated air at 7Q°F. and under atmospheric pressure 



FORMULAS FOR WORK 3 

= 14.7. For average conditions 53.5 would probably be about 
right. 

Example la. — What will be the work in foot-pounds per stroke 
done by an air compressor displacing 2 cu. ft. per stroke, com- 
pressing from p a = 14 lb. per square inch to a gage pressure 
= 701b.; compression isothermal, T = 60°F.? 

Solution (a). — Inserting the specified numerals in Eq. (1) it 
becomes 

W = 144 X 14 X 2 X log e 7 °^ 14 = 4,032 X 1.79 = 7,217. 

Solution (b). — By Tables I and II. 

By Table II the weight of a cubic foot of air at 14 lb. and 60° 
is 0.07277, and 0.07277 X 2 = 0.14554. The absolute t is 460 
+ 60 = 520, and r = 6.0. 

Then in Table I, column 11, opposite r = 6 we find 95.271, 
whence 

W = 95.271 X 520 X 0.14554 = 7,208. 

The difference in the two results is due to dropping off the fraction 
in temperature. 

Art. 2. Temperature Varying. — The conditions are said to be 
adiabatic when, during compression or expansion, no heat is al- 
lowed to enter in, or escape from, the air although the temperature 
in the body of confined air changes radically during the process. 

Physicists have proved that under adiabatic conditions the 
following relations hold: 

m = £ (3) 

P2V2 h 

and since for 1 lb. of air at 32°F. pv = 26,214 and t = 492, we 
get for 1 lb. of dry air at any pressure, volume and temperature, 

pv = 53.35f (4) 

While formulas (3) and (4) are very important, they do not apply 
to the actual conditions under which compressed air is worked, 
for in practice we get neither isothermal nor adiabatic conditions 
but something intermediate. Furthermore, moisture in the air 
will effect this coefficient. 

For such conditions physicists have discovered that the follow- 
ing holds nearly true : 

PiV! n = p x v x n = p 2 v 2 n (5) 



4 COMPRESSED AIR 

sub x indicating any intermediate stage and the exponent n 
varying between 1 and 1.41 according to the effectiveness of the 
cooling in case of compression or the heating in case of expan- 
sion. From this basic formula (5) the formulas for work must 
be derived. 

dv 
As in Art. 1, dW = p x dv x = pivi n — - n = piVi n (v x ~ n ) dv x . 

v x 

Therefore 

W = pi«i" I v x ~ n dv x = p 1 v 1 n (^ 2 — J =piv 1 n { n _ ~ — j. 

Now since p%Vi n X v 2 l ~ n = p 2 v 2 n X v 2 l ~ n = p 2 v 2 and p\V\ n V]}-~ n 
= piVi the expression becomes 

w , = P2V2 - P1V1 
n — 1 

which represents the work done in compression or expansion be- 
tween B and C (Fig. 1). To this must be added the work of ex- 
pulsion, p 2 v 2 and from it must be subtracted the work done by 
air against the back side of the piston. In case of compression 
from free air this subtraction will be p a v a - Hence, the net work 
done in one stroke of volume v a is 

T jr P2V2 — PaVa , ,„. 

W = * _ h P2V2 - PaVa (6) 

This reduces to 

71 
W = ^ZZl (V*° 2 - VaVa) (7) 

By substituting from Eq. (4), Eq. (7) may be written, for work 
in compressing 1 lb. of air, 

Wi= — ^r 53.35 (h-t a ) (7a) 

n — 1 .' 

Whenn = 1.41, W x = 183 (t 2 - t a ) (76) 

When n = 1.25, Wi = 266 (t 2 - t a ) (7c) 

Equation (7) applies also to any cases of complete expansion, 

that is, when the air is expanded until the pressure within the 

cylinder equals that against which exhaust must escape. 

Equation (7) is in convenient form for numerical computations 
and may be used when the data are in pressures and volumes, but 
it is common to express the compression, or expansion, in terms of 
r. For such cases a more convenient form of equation is gotten 
as follows: 



FORMULAS FOR WORK 

1 



T) V V n 

From Eq. (5) by factoring out one v, p 2 v 2 = 



V2 n ~ L 
Al Vl V a n ,, - V a I 

Also r = — = — -. therefore - = r» 

p a v 2 n v 2 

n—\ w — 1 n— 1 

and — — t — r n , therefore p 2 v 2 = p a v a r n 

and Eq. (7) becomes 

w-l 
W = ^PaVa(r^- l) (8) 

In cases the higher pressure, p 2 , and the less volume, v 2 , are 
known, as may sometimes be the case in complete expansion en- 
gines, we would get by a similar process 



W 



= ^ T ^[l-( 1 r )^] (8a) 

Study of the derivation of Eqs. (7) and (8) shows that they are 
equally applicable to cases of complete expansion, that is, when 
the air within the cylinder is expanded until its pressure is equal 
to that of the air outside into which the exhaust takes place. 
In ordinary cases of expansion engines apply Eq. (9). 

In perfectly adiabatic conditions n = 1.41, but in practice the 
compressor cylinders are water-jacketed and thereby part of the 
heat of compression is conducted away, so that n is less than 1.41. 
For such cases Church assumes n = 1.33 and Unwin assumes 
n = 1.25. Undoubtedly the value varies with size and propor- 
tions of cylinders, details of water-jacketing, temperature of 
cooling water and speed of compressors. Hence precision in the 
value of n is not practicable. 

For 1 lb. of air at initial temperature of 60°F. Eq. (8) gives, in 
foot-pounds, 

Whenn = 1.41, W = 95,193 (r - 29 - 1) (86) 

When n = 1.25, W = 138, 405 (r - 2 - 1) (8c) 

Common log of 95,193 = 4.978606. 

Common log of 138,405 = 5.141141. 

Values of r - 2 and r - 29 are given in Table I, columns 5 and 6 
respectively. 

. The above special values will be found convenient for approxi- 
mate computations. For compound compression see Art. 14. 

If in Eq. (8) we substitute for pv its value, 53.35 1, for 1 lb., 
we get for work on 1 lb. 



COMPRESSED AIR 

n-l 



where 



W = [(^l) 53 - 35 ( r n - 1 )] X ^ = Zi 
K =~yX 53.35 ( r ^ r - l). 



(8d) 



Note that Eq. (8cf) applies without change to cases of complete 
expansion provided that the temperature, h, of the exhaust be 
used and that r be determined to correspond, see Eqs. (12) and 
(12a). 

Table I gives values of K for n = 1.25 and n = 1.41 and for 
values of r up to 10, varying by one-tenth. The theoretic work 



*— sac= 


r 


(*)/ 


f 


e d 
c 




Fig. 2. 

in any case is K X Q X t, where Q is the number of pounds passed 
and t is the absolute initial temperature. Further explanation 
accompanies the table. 

The difference between isothermal and adiabatic compression 
(and expansion) can be very clearly shown graphically as in Fig. 
2. In this illustration the terminal points are correctly placed 
for a ratio of 5 for both the compression and expansion curve. 

Note that in the compression diagram (a) , the area between the 
two curves aef represents the work lost in compression due to 
heating, and the area between the two curves aeghb in (6) repre- 



FORMULAS FOR WORK 7 

sents the work lost by cooling during expansion. The isothermal 
curve, ae, will be the same in the two cases. 

Such illustrations can be readily adapted to show the effect of 
reheating before expansion, cooling before compression, heating 
during expansion, etc. For platting curves, see Art. 4a. 

Example 2a. — What horsepower will be required to compress 
1,000 cu. ft. of free air per minute from p a = 14.5 to a gage pres- 
sure = 80, when n = 1.25 and initial temperature = 50°F.? 

Solution. — From Table II, interpolating between 40° 
and 60° the weight of 1 cu. ft. is 0.07686 and the weight of 
1,000 is 76.86-. The r from above data is 6.5. Then in 
Table I opposite r = 6.5 in column 9 we find 0.3658. Then 



Horsepower = 0.3658 X 



76.86 
100 



X510 = 143. 



The student should check this result by Eqs. (8) or (8d) and (106) 
without the aid of the table. 

Art. 3. Incomplete Expansion. — When compressed air is applied 
in an engine as a motive power its economical use requires that it 



_Y„ J 




Fig. 3. 



be used expansively in a manner similar to the use of steam. But 
it is never practicable to expand the air down to the free air pres- 
sure, for two reasons : first, the increase of volume in the cylinders 
would increase both cost and friction more than could be balanced 
by the increase in power; and second, unless some means of re- 
heating be provided, a high ratio of expansion of compressed air 
will cause a freezing of the moisture in and about the ports. 

The ideal indicator diagram for incomplete expansion is shown 
in Fig. 3. In such diagrams it is convenient and simplifies the 
demonstrations to let the horizontal length represent volumes. 
In any cylinder the volumes are proportional to the length. 



8 COMPRESSED AIR 

Air at pressure P2 is admitted through that part of the stroke 
represented by v 2 — thence the air is expanded through the re- 
mainder of the stroke represented by v 1} the pressure dropping 
to pi. At this point the exhaust port opens and the pressure 
drops to that of the free air. The dotted portion would be 
added to the diagram if the expansion should be carried down 
to free air pressure. 

To write a formula for the work done by the air in such a case 
we will refer to Eq. (6) and its derivation. In the case of simple 
compression or complete expansion it is correctly written 

ttt V&2 — PaVa , 

W = ^ p- V&2 - PaVa, 

n — 1 

which would give work in the case represented by Fig. 1 when 
there is a change of temperature, but in such a case as is repre- 
sented by Fig. 3 the equation must be modified thus : 

w P2V2 — P1V1 . , n s 

W = ^ h P2V2 - PaVl (9) 

it A. 

the reason being apparent on inspection. 

In numerical problems under Eq. (9) there will be known p2V2,n, 
and either pi or v\. The unknown must be computed from the 
relations from Eq. (5) : 



p^p^y or v 1 = v 2 (f) 



Table I, columns 1, 2, 3 and 4, is designed to reduce the labor 
of this computation. 

Example 3a. — A compressed-air motor takes air at a gage 
pressure = 100 lb. and works with a cut-off at J4 stroke. What 
work (foot-pounds) will be gotten per cubic foot of compressed 
air, assuming free air pressure = 14.5 lb. and n = 1.41? 

Solution. — Applying Eq. (9) and noting that all pressures are to 
be multiplied by 144 and that the pressure at end of stroke 

l\£\ 1.41 

= p\ = 114.5 (y 1 ) = 16.3 and that vi = 4v 2 , we get 

/ 114.5 X 1 - 16.3 X 4 
\ 0.41 ^ 



114.5 X 1 - 14.5 X 4) = 25,444. 



FORMULAS FOR WORK 



9 



Art. 4. Work as Shown by Indicator Cards. — Volumes have 
been written on indicators and indicator cards but more than the 
following brief notes would be out of place here : 

Let li = length in inches out to out, horizontally, of indicator 
card, 
= length in feet of piston stroke, 
= spring number = pounds per inch, 
= area in square inches of indicator diagram, 
= area in square inches of piston. 

Then work per stroke is 

(10) 



I 

s 

a 

A 



W 



h 



a As 



When a planimeter is available, it is the quickest and most 
reliable means of determining the area, a, provided the operator 
know the planimeter constant. This can easily be found by run- 
ning the planimeter several times round an accurately drawn 
figure of known area, as for instance a square of 2-in. sides, and 
averaging the readings. The repetition should be made without 
lifting the tracer from the paper. It is necessary only to read 
the vernier each time the tracer reaches a fixed point on the 
figure. Then subtract each reading from the next succeeding one. 
The several differences reveal whether or not the instrument, in 
the hands of the individual can be depended on to give reliable 
results. The sum of the differences divided by the number of 





Vernier 


Difference 


Error 


Per cent. 


Reading at start 


794 
184 
578 
964 
362 
753 


390 
394 
386 
398 
391 


1 

392 

3 
392 

5 
392 

7 
392 


392 




Pass zero 


-0.25 


After first round 




After second round 


-0.75 


After third round 


-1.28 


Pass zero 

After fourth round 


-1.80 


After fifth round 


0.00 







Total for five rounds, 1,959 

1 959 4 00 

' r = 392 = average and o^ = 1-03 = planimeter constant. 



10 



COMPRESSED AIR 



repetitions gives the average and the average divided into the 
known area gives the planimeter constant. 

Example. — The table, page 9, shows records and deductions for 
a planimeter tested on a rectangle of 4 sq. in.: 

To get the full information shown by an indicator diagram 
taken from an air compressor, there should be placed on it the 
clearance line and the isothermal curve. For direct determina- 
tion of clearance see Art. 46. 




A 10 



Fig. 4. 



Referring to Fig. 4, the clearance line, OC, is placed at a dis- 

DO 

tance, DO, such that -ry. = percentage of clearance. If clear- 
ance has been determined by measurements in the machine, the 
line OC is set out by measuring a distance CB as determined above. 
If clearance has not been measured in the machine, the position 
of the line OC or point can be computed as follows : 

Scale one of the pressure ordinates where the curve is smooth. 
Represent this by p x ; the distance from its foot to the unknown 
point represent by x and the known distance from A to the 
foot of p x by k. 



FORMULAS FOR WORK 11 

Then by Eq. (5) 

p a + k) n = p x x n or x + k = l—) n i 



1 

l X. 



\p 

Whence x =- j 

(P±)n- 1 
\pj 



(a) 



This method is of course dependent on the assumed value of n. 

By the same principle, if can be correctly located independ- 
ently of n, the value of n can be computed thus: x is now known. 
So let x -f- k = I. 

rpi • I" Px j log Px - l Og Pa m 

Then — = — and n = ° 7 , (o) 

rc n p a log £ — log # 

In large well-designed air compressors the clearance should not 
exceed 1 per cent. Then OC would very nearly coincide with 
DB but would always be a little outside. 

Note that if x from Eq. (a) places inside of D, it is evidence 
that n has been assumed too small. 

In many books on steam engines and air compressors can be 
found instructions for locating the point graphically. Con- 
cerning these, the student is warned that they are all based on the 
assumption that the curve is the isothermal; and hence are apt 
to give very misleading results. For instance, one method is to 
construct a rectangle on the curve as developed by the indicator, 
as shown in mnqr, and the diagonal nr will pass through 0. 
This would be correct for the isothermal EF but evidently not so 
for the actual curve EG. 

Before passing, the student's attention should be directed to the 
fact that in steam engines the clearance is usually much greater 
than is allowed in air compressors. In steam engines it varies 
much and sometimes goes to 10 or even 15 per cent. ; and further, 
the curves of steam-engine indicator cards are much more erratic 
than for air compressors, due to condensation during expansion 
and compression without cooling, which may cause reevapora- 
tion. 

After all is said, if the investigator wants to know what the 
clearance is he should measure it in the machine. 

The curves, either isothermal or adiabatic, can most readily 
be set out by dividing the line OA into ten equal parts numbered 
as shown, then letting x = number of divisions from the pres- 
sure ordinate at the xth division will be, for the isothermal curve : 



12 COMPRESSED AIR 



Px = 



and for the adiabatic curve p x = p a ( — ) 



x 

The following table applies where p a — 14.7: 

x = 10 9876543 2 

Isothermal p x = 14.7 16.3 18.2 21.0 24.5 29.4 36.6 49.0 78.5 
Adiabatic p x = 14.7 17.1 20.1 23.5 29.3 39.1 53.5 80.3 142.1 

The isothermal curve is symmetrical about the middle line 
and the upper half can be set out from the other axis OB with the 
same ordinates used to plat the first half. 

Art. 4a. Mean Effect Pressures.— In much of the literature 
relating to work done in steam engines and air compressors, use is 
made of the term "mean effective pressure" — abbreviated m.e.p. 
A definition of the term is: A pressure which multiplied by the 
volume, or piston displacement, gives work. 

Then to find the m.e.p. in compression when the volume is v 
we have: 

In isothermal compression (or expansion) 

(m.e.p.)v = p a v log e r 
and m.e.p. = p a log e r = 2.3p a logio r. (10) 

In case of adiabatic compression (or complete expansion) 

(n — 1 . 

r n — lj 

(n — 1 . 

r n — lj (10a) 

n — 1 
Values ofr - _ are given in Table I, column 5, for n = 1.25. 

n 

In case of incomplete expansion (Eq. 9) when the cutoff is at 
k per cent, of the stroke, or v 2 = kv. 

, v p 2 kv — piv 7 

(m.e.p.) v = — f [- p 2 kv — p a v. 

lb J. 

From the condition that p\V n = p%vi n , we get 
whence the above equation reduces to 



p 2 (kn — k n ) (106) 

— --> — - - Pa 

n — 1 

To reduce computations of m.e.p. in this case to simple arith- 



m.e.p. = - — ^j p a 

n — 1 



FORMULAS FOR WORK 



13 



metic, the values of 



kn — k n 



are given below for n = 1.25 and for 



n — 1 

a sufficient range in k to meet the demands of ordinary practice. 
These values will apply to any gas, including steam so long as 
there is no condensation of the steam. 



f Fraction 
{_ Decimal 
kn - k n 



0.1250 
0.3287 



0.1875 
0.4439 



H 

0.2500 

0.5428 



5ie 
0.3125 

0.6281 



3 A 
0.3750 

0.7006 



yi6 

0.4375 
0.7643 



0.5000 
0.8180 



?18 

0.5625 
0.8641 



0.6250 
0.9022 



Art. 4b. Effect of Clearance in Compression. — It is not prac- 
ticable to discharge all of the air that is trapped in the cylinder. 
There are some pockets about the valves that the piston cannot 
enter, and the piston must not be allowed to strike the head of the 
cylinder. This clearance can usually be determined by measuring 
the water that can be let into the cylinder in front of the piston 
when at the end of its stroke; but the construction of each com- 
pressor must be studied before this can be undertaken intelli- 
gently, and it is not done with equal ease in all machines. 

To formulate the effect of this clearance in the operation of 
the machine, 

Let v = volume of piston displacement (= area of piston X 
length of stroke), 

Let cv = clearance, c being a percentage. 

Then v + cv is the volume compressed each stroke. But 

i 
the clearance volume cv will expand to a volume r n cv as the 
piston recedes, so that the fresh air taken in at each stroke will 

'i 
be v + cv — r n cv, and the volumetric efficiency will be 



E v = 



v + cv 



i 

r n cv 



= 1 + c (1 - r n ) 



(ID 



Theoretically (as the word is usually used) clearance does not 
cause a loss of work, but practically it does, insomuch as it 
requires a larger machine, with its greater friction, to do a given 
amount of effective work. 

Example 46. — A compressor cylinder is 12-in. diameter by 
16-in. stroke. The clearance is found to hold 134 pt- of water 

= -~ X 231 = 36 cu. in., therefore c = ^^ v 1fi = 0.02. 



14 



COMPRESSED AIR 



Then by Eq. (11) when r = 7 and n = 1.25. 

E = 1 '+ 0.02 (1 - 7 - 8 ) = 92 per cent. 

Such a condition is not abnormal in small compressors, and the 
volumetric efficiency is further reduced by the heating of air 
during admission as considered in Art. 6. 

Art. 5. Effect of Clearance and Compression in Expansion 
Engines. — Figure 5 is an ideal indicator diagram illustrating the 
effect of clearance and compression in an expansion engine. 

In this diagram the area E shows the effective work, D the 
effect of clearance, B the effect of back pressure of the atmosphere 
and C the effect of compression on the return stroke. 

The study of effect of clearance in an expansion engine differs 
from the study of that in compression, due to the fact that the 




b — bl— ^ 



Fig. 5. 



volume in the clearance space is exhausted into the atmosphere 
at the end of each stroke. 

If the engine takes full pressure throughout the stroke the air 
(or steam) in the clearance is entirely wasted; but when the air is 
allowed to expand as illustrated in the diagram some useful work 
is gotten out of the air in the clearance during the expansion. 

The loss due to clearance in such engine is modified by the 
amount of compression allowed in the back stroke. If the com- 
pression p c = p 2 , the loss of work due to clearance will be noth- 
ing, but the effective work of the engine will be considerably 
reduced, as will be apparent by a study of a diagram modified to 
conform to the assumption. 

While the formula for work that includes the effect of clear- 
ance and compression will not be often used in practice its deriva- 
tion is instructive and gives a clear insight into these effects. 



FORMULAS FOR WORK 15 

The symbols are placed on the diagram and will not need fur- 
ther definition. 

The effective work E will be gotten by subtracting from the 
whole area the separate areas B, C and D. From Art. 2, after 
making the proper substitutions for the volumes, there results 

m x 1 ,rP2(c + fc) — pi(l+c) , / , 7 sl 

Total area = l\ — _ i h y% (c +.&) . 



Area 5 = Zp a , 
Area D = lp 2 c, 



Area C 



= l[ W ~ * <» + 3 _ pa5 ] 



Subtracting the last three from the first and reducing their 

results : 

Work 1 

—j^ = ^ZT\ t c (Pa + Pa - 2? c - Pi) + n (p 2 & + pJ>-p a ) - (pi 

— p )] = mean effective pressure. 
The actual volume ratio before and after expansion is 

V2 _ c?i + kit _ c + k 

V\ eh + li c + 1 

This is the ratio with which to enter Table I to get r and t and 
from r the unknown pressure pi. Similarly, for the compression 

curve the ratio of volumes is r> and p c can be found as indicated 

above. 

Art. 5a. Adjustment of Mechanically Operated Intake 
Valves. — While no attempt will be made to show details of valves, 
it is appropriate to call attention here to the fact that discharge 
valves to an air compressor are nearly always of the "poppet" 
type, that is they pop open automatically when the pressure in- 
side the compressor exceeds that in the receiver. Thus the time, 
or point of opening of the discharge valve, will adjust itself to any 
variation of pressure in the receiver, a condition evidently desir- 
able. But the intake valves may be mechanically operated, and 
are so operated in many of the larger machines. When so oper- 
ated, the machine works more efficiently, with less noise and there 
is less liability to breakdown. 

The correct adjustment of the point of opening of mechanically 
operated inlet valves depends on the clearance and the ratio of 
compression. 



16 



COMPRESSED AIR 



Figure 5a illustrates one class of inlet valve with its operating 
mechanism, the direction of motion being indicated by arrows. 
The piston d is at the left end of its stroke with compressed air 
in the clearance. If the port of valve A opens at this instant, the 
compressed air in the clearance will escape out into the inlet 
passage e. Evidently it is desirable to delay the opening of the 
port until the piston has receded enough to allow the air in the 
clearance to expand down to atmospheric pressure, thus letting 
the air in the clearance give back the work done in compressing 
it. Evidently, the opening should not be delayed longer, for 
there would result a suction (pressure below atmosphere) behind 
the piston which would cause a loss of work. When the adjust- 
ment is correct, there is no puffing or spitting at the inlet parts. 
When the adjustments are not correct, an experienced operator 
can detect the fact by the noise made by air puffing out or into 
the parts. 




SECTION .MN" 



SECTION QK 



Fig. 5a. 



If the clearance and the ratio of compression are known, the 
erector or operator can adjust the valves correctly. For example, 
assume the clearance as 1 per cent., r = 7 and n = 1.25. In 
Table I for r = 7, v 2 -=- Vi = 0.21 or say vi = 5v 2) that is the clear- 
ance should expand to five times its volume before the port opens. 
Otherwise stated, the piston should move back 4 per cent, of its 
stroke before the port opens. Thus, if the stroke be 18 in. the 
piston should be moved back 0.04 X 18 = 0.72 (or say Y± in.) 
and while the piston stands in that position bring the edge of 
valve and edge of inlet port to coincide by turning the rod 6 or a 
as the case may be. The manufacturers always put marks on 
the end of the valve and on the inclosing cylinder that will enable 
the operator to make this adjustment. 

In order that one adjustment may not interfere with another, 
it is necessary that the valve B be adjusted first by rotating 



FORMULAS FOR WORK 17 

rod a; then adjust valve A by rotating rod 6. If the compressor 
be a compound tandem, adjust the valves in the order of their 
distance from the eccentric. 

Art. 6. Effect of Heating Air as it Enters Cylinders. — When 
a compressor is in operation all the metal exposed to the compressed 
air becomes hot even though the water-jacketing is of the best. 
The entering air comes into contact with the admission valves, 
cylinder head and walls and the piston head and piston rod, and 
is thereby heated to a very considerable degree. In being so 
heated the volume is increased in direct proportion to the absolute 
temperature (see Eq. (3)), since the pressure may be assumed 
constant and equal that of the atmosphere. Hence a volume of 
cool free air less than the cylinder volume will fill it when heated. 
This condition is expressed by the ratio 

V a "a "a 

- ■ = — or V a = Vc -> 

Vc Zc tc 

where v c and t c represent the cylinder volume and temperature. 
The volumetric efficiency as effected by the heating is 

V c t c 

Example 6. — Suppose in Ex. 4a the outside free air tempera- 
ture is 60°F. and in entering the temperature rises to 160°F., 
then 

t a 460 + 60 .. , 

^ = 460 + 160 = 84perCent ' 

Then the final volumetric efficiency would be 92 X 84 = 77 
per cent, nearly. 

The volumetric efficiency of a compressor may be further re- 
duced by leaky valves and piston. 

In Arts. 4& and 6 it is made evident that the volumetric efficiency 
of an air compressor is a matter that cannot be neglected in any 
case where an installation is to be intelligently proportioned. It 
should be noted that the volumetric efficiency varies with the 
various makes and sizes of compressors and that the catalog 
volume rating is always based on the piston displacement. 

These facts lead to the conclusion that much of the uncertainty 
of computations in compressed-air problems and the conflict- 
ing data recorded is due to the failure to determine the actual 



18 COMPRESSED AIR 

amount of air involved either in terms of net volume and tempera- 
ture or in pounds. 

Methods of determining volumetric efficiency of air compres- 
sors are given in Chapter II. 

The loss of work due to the air heating as it enters the com- 
pressor cylinder is in direct proportion to the loss of volumetric 
efficiency due to this cause. In Ex. 6a only 84 per cent, of the 
work done on the air is effective. 

By the same law any cooling of the air before entering the 
compressor effects a saving of power. See Art. 10. 

Art. 7. Change of Temperature in Compression or Expan- 
sion. — Equation (4) may be written for any fixed weight of air 

PlVl = Ctl) 2?2^2 = cto 

and Eq. (5) may be factored thus, 

P\ViVi n ~ l = J>lV 2 V 2 n ~ l . 

Substituting we get 

CtiVi n ~ l = ct 2 v 2 n ~ 1 . 

Whence U = *i(^)"~ X (12) 

and . t 2 = *i(— )~"~ = tif»> (12a) 

i 
since from Eq. (5) — = ( — ) n - 

It is possible to compute n from Eq. (12) by controlling the V\ and 
v 2 and measuring t\ and t 2 . 

Table I, columns 5 and 6, is made up from Eq. (12a) and 
columns 3 and 4 from Eq. (5) as just written. 

Example 7. — What would be the temperature of air at the end 
of stroke when r = 7 and initial temperature = 70°F. ? 

Solution. — Referring to Table I in line with r = 7 note that 

1.4758 when n = 1.25 

:.h= (460 + 70) X 1.4758 - 460 = 322°F. 
1.7585 when n = 1.41 

:.t%= (460 4- 70) X 1.7585 - 460 = 472°F. 

From the same table the volume of 1 cu. ft. of free air when 
compressed and still hot would be respectively 0.21 and 0.25, 



FORMULAS FOR WORK 19 

while when the compressed air is cooled back to 70° its volume 
would be 0.143. 

Art. 8. Density at Given Temperature and Pressure. — By Eq. 
(4) pv = 53.35 for 1 lb., and the weight of 1 cu. ft. = 1 lb. divided 
by the volume of 1 lb. 

Therefore w = ] = ^ (13) 

Note that p must be the absolute pressure in pounds per square 
foot, and t the absolute temperature. When gage pressures are 
used and ordinary Fahrenheit temperature the formula becomes 

w- 144 /^ + ?M 



53.35 



V460 + Fl 



Table III is made up from Eq. (13). 

Art. 8a. Weight of Moist Air. — In some cases where unusual 
refinement of calculations may be required it will be necessary to 
take cognizance of the fact that air containing water vapor is not 
equal in weight to pure air at the same pressure and temperature. 
In case of moving air at atmospheric pressure as in case of fans 
and blowers and where air is measured at atmospheric pressure by 
means of orifices, the error resulting from neglecting moisture may 
be as much as l}^ per cent. 

Since atmospheric pressure and water-vapor pressures are 
usually recorded in inches of mercury it will be convenient to re- 
tain in the formulas inches head of mercury instead of pounds per 
square inch. 

Let m = pressure (absolute) of mixture of air and water vapor 
in inches of mercury, 
q = pressure of saturated water vapor at given tempera- 
ture in inches of mercury (to be found in steam 
tables), 
H = percentage of humidity, 

K = ratio of weight of water vapor to dry air at given 
temperature, 
t = absolute temperature = 459.6 + F. 

To adopt Eq. (13), viz., W a = p .4- 53.35 £ to this case, note 
that 2.036 in. of mercury gives 1 lb. pressure and that Hq ( = 



20 COMPRESSED AIR 

vapor pressure at H humidity) must be subtracted from p in order 
to get the true pressure of the air. 

_ 144 (m - Hq) 1.3253 (m - Hq) 
men w a 2m& x ^^ - f 

and weight of water vapor in a cubic foot is 

1.3253 VTJ 
w w = — - — KHq. 

Then the combined weight is 

w a + w w = w = - 1 — — [m — Hq (1 — K)] (13&) 

t 

For values of K see Table Ilia, page 134, which is copied from 
Engineering News, June 18, 1908, or Compressed Air Magazine, 
vol. 13, p. 4967. These articles give also a very full treatment 
of the subject of moisture in air. 

The ratio K varies between 0.611 at zero degrees F. and 0.623 
at 100°F. Some writers assume it constant. If we assume it 
constant and equal 0.62 (which is correct for temperature 74°), 
then the equation becomes 

w = ^^ (w - 0.38tfg) (13c) 

Example 13c. — Find the weight per cubic foot of air in a duct 
leading away from a fan when T = 70°F. Barometer reading 
in free air = 28.85-in.-water gage, (i), = 4 in. and humidity, (H), 
= 80 per cent. 

Solution. — 4 in. of water = 0.29 in. of mercury. Then m 
= 29.14. 

At 70° K = 0.6196 and 1 - K = 0.3804. 

At 70° q = 0.739 and Hq = 0.5912. 

1 QQ'lS 

Then w = -~^ (29.14 - 0.5912 X 0.3804) = 0.07233. 

Pure air under the same pressure and temperature would have 
w a = 0.07287, a difference of less than 1 per cent. If the air were 
saturated the difference would be greater. 

Art. 9. Cooling Water Required. — In isothermal changes, since 
pv is constant, evidently there is no change in the mechanical 
energy in the body of air as measured by the absolute pressure and 
using the term " mechanical energy" to distinguish from heat 
energy. Hence evidently all the work delivered to the air from 



FORMULAS FOR WORK 21 

outside must be abstracted from the air in some other form, and 
we find it in the heat absorbed by the cooling water. Therefore, 

pv log e r 



778 



= (B.t.u.'s) 



of work must be absorbed by the cooling water. If the water 
is to have a rise of temperature T° (T being small, else the 
assumption of isothermal changes will not hold), then 

Pv\oS e V 

7R0 T = P oun< ^ s °f water required in same time. 

Example. 9 — How many cubic feet of water per minute will be 
required to cool 1,000 cu. ft. of free air per minute, air compressed 
from p a = 14.2 to p g = 90° gage, initial temperature of air = 
50°F. and rise in temperature of cooling water = 25°? 

Solution. — 

/ 90 -4- 14 2 \ 
144 X 14.2 X 1,000 X log e ( ^ g ) 

780 X 25 X 62.5 = 3 " 36 CU " ft per 

minute. 

It is practically possible to attain nearly isothermal conditions 
by spraying cool water into the cylinder during compression. In 
such a case this article would enable the designer to compute the 
quantity of water necessary and therefrom the sizes of pipes, 
pumps, valves, etc. 

Art. 10. Reheating and Cooling. — In any two cases of change 
of state of a given weight of air, assuming the ratio of change in 
pressure to be the same, the work done (in compression or expan- 
sion) will be directly proportional to the volume, as will be evident 
by examination of the formulas for work. Also, at any given 
pressure the volumes will be directly proportional to the absolute 
temperatures. Hence the work done either in compression 
or expansion (ratio of change in pressures being the same in 
each case) will be directly p: portional to the absolute initial 
temperatures. 

Thus if the temperature of the air in the intake end of one com- 
pressor is 150°F. and, in another 50°F., the work done on equal 
weights of air in the two cases will be in the proportion of 460 + 
150 to 460 + 50, or 1.2 to 1; that is, the work in the first case is 
20 per cent, more than that in the second case. This is equally 
true, of course, for expansion. 



22 COMPRESSED AIR 

The facts above stated reveal a possible and quite practicable 
means of great economy of power in compressing air and in using 
compressed air. 

The opportunities for economy by cooling for compression are 
not as good as in heating before the application in a motor, but 
even in compression marked economy can be gotten at almost 
no cost by admitting air to the compressor from the coolest con- 
venient source, and by the most thorough water-jacketing with 
the coolest water that can be conveniently obtained. 

In all properly designed compressor installations the air is 
supplied to the machine through a pipe from outside the building 
to avoid the warm air of the engine room. In winter the differ- 
ence in temperature may exceed 100°, and this simple device 
would reduce the work of compression by about 20 per cent. 



Gas, Liquid or -y j 
Powdered Fuel — *" I — » 



L 

> A Hot Air 



Combustion 

JLJLfi_sx jUSLSLJSLju 



Fig. 6. 

For the effect of intercoolers and interheaters see Art. 11 on 
compounding. 

By reheating before admitting air to a compressed-air engine 
of any kind the possibilities of effecting economy of power are 
greater than in cooling for compression, the reason being that 
heating devices are simpler and less costly than any means of 
cooling other than those cited above. 

The compressed air passing to an engine can be heated to any 
desired temperature; the only limit is that temperature that will 
destroy the lubrication. Suppose the normal temperature of the 
air in the pipe system is 60°F. and that this is heated to 300°F. 
before entering the air engine, then the power is increased 46 
per cent. Reheating has the further advantage that it makes 
possible a greater ratio of expansion without the temperature 
reaching freezing point. 

The devices for reheating are usually a coil or cluster of pipes 
through which the air passes while the pipe is exposed to the heat 



FORMULAS FOR WORK 23 

of combustion from outside. Ordinary steam boilers may be 
used, the air taking the place of the steam and water. 

Unwin suggests reheating the air by burning the fuel in the 
compressed air as suggested in the cut. 

Even when the details are worked out such a device would be 
simple and inexpensive. The theoretic advantages of such a 
device are that all the heat would go into the air, the gases of com- 
bustion (if solid or liquid fuel be used) would increase the volume, 
and the combustion occurring in compressed air would be very 
complete. 

The author has no knowledge of any such devices having been 
used in practice. 1 

The power efficiency of the fuel used in reheaters is very much 
greater than that of the fuel used in steam boilers. Unwin states 
that it is five or six times as much. The chief reason is that none 
of the heat is absorbed in evaporation as in a steam boiler. 

In many of the applications of compressed air reheating is im- 
practicable, and efficiency is secondary to convenience — but in 
large fixed installations, such as mine pumps, reheating should be 
applied. 

Art. 11. Compounding. — In steam-engine designs compound- 
ing is resorted to to economize power by saving steam, while in air 
compressors and compressed-air engines compounding is resorted 
to for the twofold purpose of economizing power and controlling 
temperature, both objects being accomplished by reducing the 
extreme change of temperature. The economic principles in- 
volved in compound steam engines and in compound air engines 
are quite different, the reasons underlying the latter being much 
more definite. 

The air is first compressed to a moderate ratio in the low- 
pressure cylinder, whence it is discharged into the "inter cooler," 
where most of the heat developed in the first stage is absorbed and 
thereby the volume materially reduced, so that in the second 
stage there will be less volume to compress and a less injurious 
temperature. 

The changes occurring and the manner in which economy is 

1 Since the publication of the first edition a very promising device has 
appeared in which the current of compressed air automatically injects the 
fuel oil; thus, presumably, maintaining a constant proportion between the 
quantity of air and of oil, so that the temperature of the discharged air will 
be constant. 



24 



COMPRESSED AIR 



effected in compression may be most easily understood by refer- 
ence to Fig. 7, which represents ideal indicator diagrams from 
the two cylinders, superimposed one over the other, the scale 
being the same in each, the dividing line being kb. 

In this diagram, 
abk is the compression line in the first-stage or low-pressure 

cylinder, 
cds is the compression line in the second-stage or high-pressure 

cylinder, 
be is the reduction of volume in the intercooler, with pressure 

constant, 



e d f g 




Fig. 7. 



abf would be the pressure line if no intercooling occurred, 
The area cdfb is the work saved by the intercooler, 
ace would be the compression line for isothermal compression, 
aug would be the compression line for adiabatic compression. 

The diagram is correctly proportioned for r = 6. 

Figure 8 is a diagram drawn in a manner similar to that used 
in Fig. 7 and is to illustrate the changes and economy effected 
by compounding with heating when compressed air is applied in 
an engine. It is assumed that the air is "preheated," that is, 
heated once before entering the high-pressure cylinder and again 
heated between the two cylinders. 



FORMULAS FOR WORK 



25 



In this diagram, 
se is the volume of compressed air at normal temperature, 
sf is the volume of compressed air after preheating, 
fc is the expansion line in the high-pressure cylinder, 
cb is the increase of volume in the interheater, 
by is the expansion line in low-pressure cylinder, 
ezq would be the adiabatic expansion line without any heating, 
efcz is work gained by preheating, 
cbyx is work gained by interheating. 

In no case is it economical to expand down to atmospheric 
pressure. Hence the diagram is shown cut off with pressure still 
above that of free air. 





s 


e f 






\ * 

V \ 

V » 

V » 
V v 

V * 

\\ 


\ » * 

\ \ * 


p 


s 








] 


> 
1 

Pa 

, 4 


Pa "■"■- 

i 


y 







Fig. 8. 



The diagram, Fig. 8, is proportioned for preheating and reheat- 
ing 300°F. 

Art. 12. Proportions for Compounding. — It is desirable that 
equal work be done in each stage of compounding. If this con- 
dition be imposed, Eq. (8) indicates that the r must be the same 
in each stage, for on the assumption of complete intercooling the 
product pv will be the same at the beginning of each stage. 

If then ri be the ratio of compression in the first stage, the 
pressure at end of first stage will be rip = p h and the pressure at 
end of second stage = npi = ri 2 p a = p%, and similarly at end 
of third stage the pressure will be p 3 = ri z p a , or 



26 COMPRESSED AIR 



In two T stage work r\ = (— Y = r 2 1/i# 
In three-stage work r x = ( — | ^ = r 3 * 



vp 

Let yi = free air intake per stroke in low-pressure cylinder or first 
stage, 
v 2 = piston displacement in second stage, 
v s = piston displacement in third stage, 
r.i = ratio of compression in each cylinder. 

Then, assuming complete intercooling, 

Vi , V 2 Vi 

v 2 = ~ and v s = — = — -j 

or 

IH 1 ,V 3 1 

— = — and — = — 5* 
vi rx vi r x 2 

The length of stroke will be the same in each cylinder; there- 
fore the volumes are in the ratio of the squares of diameters, or 

dl _ 1. A dl _ 1_ 

dji - ^ allCl di 2 ~ ri 2" 



Hence 



d 2 = -Aj and d z = — (14) 

ri /2 r x 



If the intention to make the work equal in the different cylin- 
ders be strictly carried out it will be necessary to make the first- 
stage cylinder enough larger to counteract the effect of volu- 
metric efficiency. Thus if volumetric efficiency be 75 per cent., 
the volume (or area) of the intake cylinder should be one-third 
larger. Note that the volumetric efficiency is chargeable 
entirely to the intake or low-pressure cylinder. Once the air is 
caught in that cylinder it must go on. 

Example 12. — Proportion the cylinders of a compound two- 
stage compressor to deliver 300 cu. ft. of free air per minute at a 
gage pressure = 150. Free air pressure = 14.0, r.p.m. = 100, 
stroke 18 in., piston rod 1% in. diameter, volumetric efficiency 
= 75 per cent. 

Solution. — From the above data the net intake must be 3 cu. ft. 
per revolution. Add to this the volume of one piston rod stroke 
( = 0.025 cu. ft.), and divide by 2. This gives the volume of one 
piston stroke 1.512. The volume of 1 ft. of the cylinder will be 



FORMULAS FOR WORK 27 

12 

jg X 1.512 = 1.008 cu. ft. From Table X the nearest cylin- 
der is 14 in. in diameter, the total ratio of compression = 

77 = 11.71, and the ratio in each stage is (11.71) I/§ = 

3.7 = r h and by (14) 

v di 14 _ . _ 

= (nP = L92 = m '' say A lrlv 

for the high-pressure cylinder. 

Now we must increase the low-pressure cylinder by one-third to 
allow for volumetric efficiency. The volume per foot will then 
be 1.344, which will require a cylinder about 15% in. in diame- 
ter. Note that the diameter of the high-pressure cylinder will 
not be affected by the volumetric efficiency. 

Art. 13. Work in Compound Compression. — Assuming that 
the work is the same in each stage, Eq. (8) can be adapted to the 
case of multistage compression thus : 

In two-stage work 

... 

n-l 1 

= ^T- l VaVa(r^- l) X 2. (15a) 

In three-stage work 

n-l 

W = ~j p a v a (r^ - l) X 3 (16) 



W -^VaVain "-1 X2 (15) 



n — 1 



n-l 

V 



n— i 

v a (r, *"" - l) X 3 (16a) 



Note that r 2 = — and r 3 = — and also that p a v a = ViVi = 

Pa Pa 

p 2 v 2 , etc., assuming complete intercooling. 

Laborious precision in computing the work done on or by com- 
pressed air is useless, for there are many uncertain and changing 
factors; n is always uncertain and changes with the amount and 
temperature of the jacket water, the volumetric efficiency, or 
actual amount of air compressed, is usually unknown, the value 
of p a varies with the altitude, and r is dependent on p a . 

Art. 14. Work under Variable Intake Pressure. — There are 
some cases where air compressors operate on air in which the in- 



28 COMPRESSED AIR 

take pressure varies and the delivery pressure is constant. This 
is true in case of exhaust pumps taking air out of some closed 
vessels and delivering it into the atmosphere. It is also the con- 
dition in the "return-air" pumping system in which one charge 
of air is alternately forced into a tank to drive the water out and 
then exhausted from the tank to admit water. For full mathe- 
matical discussion of this pump see Trans. Am. Soc. C. E., vol. 
54, p. 19. The formulas of Arts. 14 and 15 were first worked 
out to apply to that pumping system. 

In such cases it is necessary to determine the maximum rate of 
work in order to design the motive power. 

First assume the operation as being isothermal. Then in Eq. 
(1), viz., 

W = p x v\og e ~^> 
Px 

p x is variable, while v and py are constant. In this formula W 
becomes zero when p x is zero and again when p x = pi, since log 
1 is zero. To find when the work is maximum, differentiate and 
equate to zero; thus differential of 

v (p x log e px - Px log e p x ) = v |^log e pidp x - (p x ^ + log, p x dp x .J J • 

Equate this to zero and get log e pi = 1 + log e p X) 
or 

log e — = 1, therefore ^ = e = 2.72. 

Px Px 

That is, when r = 2.72 the work is a maximum. 

When the temperature exponent n is to be considered the study 
must be made in Eq. (8), viz. 



W = p x v 

71—1 



(21)— -1 



(8) 



■Pi 
Differentiating this with respect to p x and equating to zero, 

w-l 

the condition for maximum work becomes ( -j j r = n. Insert 
this in (8) and the reduced formula becomes 

TXT VlV 

W = np x v. = — — 



FORMULAS FOR WORK 29 

From the above expression for maximum the following results : 

When n = 1.41 the maximum occurs when r = 3.26. 

When n = 1.25 the maximum occurs when r = 3.05. 

When n = 1.00 the maximum occurs when r = 2.72. 
In practice r = 3 will be a safe and convenient rule. 

Exercise 14a. — Air is being exhausted out of a tank by an ex- 
haust pump with capacity = 1 cu. ft. per stroke. At the begin- 
ning the pressure in the tank is that of the atmosphere = 14.7 
lb. per sq. in. Assume the pressure to drop by intervals of 1 lb. 
and plot the curve of work with p x as the horizontal ordinate and 
W as the vertical, using the formula 

W = p x vlog e —- 

Vx 

Exercise 146. — As in 14a plot the curve by Eq. (8) with n = 
1.25. 

Art. 15. Exhaust Pumps. — In designing exhaust pumps the 
following problems may arise. 

Given a closed tank and pipe system of volume V under pres- 
sure po and an exhaust pump of stroke volume v, how many 
strokes will be necessary to bring the pressure down to p m ? 

The analytic solution is as follows, assuming isothermal condi- 
tions in the volume V. 

The initial product of pressure by volume is poV. After the 
first stroke of the exhaust pump this air has expanded into the 
cylinder of the pump and pressure has dropped to pi. Under the 
law that pressure by volume is constant; 

(V + v) pi = p V, or p! = ^r v 

at end of first stroke, 

PiV l V 

at end of second stroke, 

V ( V 

at end of third stroke, etc. 
Finally 



ViV I V \ 2 

(V + v)p 2 = Vl V, or p 2 = y-^ = v*\Y+r v ) 

f second stroke, 

V I V \ 

(V + v) p z = p 2 V, or p 3 = p 2 y^±T v = P° \v^f~v) 



1 Vr 

log r 



I V \ m , & p 

log (v + y 



30 COMPRESSED AIR 

This is inconvenient for solution on account of the minus charac- 
teristics. Hence it is better to write it thus: 

_ log p m - log p . 

fro — 



1.16136 




log 101 


= 2.00432 


0.69897 




log 100 


= 2.00000 


0.46239 






0.00432 


46239 
432 


= 107 


= m. 





log V - log (V + v) 
Now change sign of both numerator and denominator and we get 

» = lo , s /°-, log f- (17) 

log ( V + v) — log V 

Example 15a. — A closed tank containing 100 cu. ft. of air at 
atmospheric pressure (14.7 lb.) is to be exhausted down to 5 lb. 
by a pump making 1 cu. ft. per stroke. How many strokes are 
required? 

Solution.— po = 14.7, p m ' = 5, V + v = 101 and V = 100. 



log 5 



The results found under Arts. 14 and 15 serve well to illustrate 
the curious mathematical gymnastics that compressed air is sub- 
ject to, and should encourage the investigator who likes such 
work, and should put the designer on guard. 

Art. 16. Efficiency when Air is Used without Expansion. — In 
many applications of compressed air convenience and reliability 
are the prime requisites, so that power efficiency receives little 
attention at the place of application. This is so with such appara- 
tus as rock drills, pneumatic hammers air hoists and the like. 
The economy of such devices is so great in replacing human labor 
that the cost in power is little thought of. Further, the necessity 
of simplicity and portability in such apparatus would bar the 
complications needed to use the air expansively. There are other 
cases, however, notably in pumping engines and devices of vari- 
ous kinds, where the plant is fixed, the consumption of air con- 
siderable and the work continuous, where neglect to work the air 
expansively may not be justified. 

In any case the designer or purchaser of a compressed-air plant 
should know what is the sacrifice for simplicity or low first cost 
when the proposition is to use the air at full pressure throughout 
the stroke and then'exhaust the cylinder full of compressed air. 



FORMULAS FOR WORK 31 

Let p be the absolute pressure on the driving side of the piston 
and p a be that of the atmosphere on the side next the exhaust. 
Then the effective pressure is p — p a and the effective work is 
(p ~ Pa) v, while the least possible work required to compress this 
air is pv log e r. 

Hence the efficiency is 

E = (P ~ Pa) V 

pv loge r 
Dividing numerator and denominator by p a v this reduces to 

T — 1 

E = ~ — - (18) 

r loge r 

This is the absolute limit to the efficiency when air is used without 
expansion and without reheating. It cannot be reached in 
practice. 

Table VI represents this formula. Note that the efficiency de- 
creases as r increases. Hence it may be justifiable to use low- 
pressure air without expansion when it would not be if the air 
must be used at high pressure. 

Clearance in a machine of this kind is just that much com- 
pressed air wasted. If clearance be considered, Eq. (18) becomes 

E = . T ~ \ (18a) 

(1 + c) r loge r 

where c is the percentage of clearance. In some machines, if 
this loss were a visible leak, it would not be tolerated. 

Art. 17. Variation of Atmospheric Pressure with Altitude. — 
In most of the formulas relating to compressed-air operations the 
pressure p a , or weight w a , of free air is a factor. This factor varies 
slightly at any fixed place, as indicated by barometer readings, 
and it varies materially with varying elevations. 

To be precise in computations of work or of weights or volumes 
of air moved, the factors p a and w a should be determined for each 
experiment or test, but such precision is seldom warranted further 
than to get the value of p a for the particular locality for ordinary 
atmospheric conditions. This, however, should always be done. 
It is a simple matter and does not increase the labor of computa- 
tion. In many plants in the elevated region p a may be less than 
14.0 lb. per square inch, and to assume it 14.7 would involve an 
error of more than 5 per cent. 

Direct reading of a barometer is the easiest and usual way of 



32 COMPRESSED AIR 

getting atmospheric pressure, but barometers of the aneroid class 
should be used with caution. Some are found quite reliable, but 
others are not. In any case they should be checked by compari- 
son with a mercurial barometer as frequently as possible. 

If m be the barometer reading in inches of mercury and F be 
the temperature (Fahrenheit), the pressure in pounds per square 
inch is 

p a = 0.4912 m[l - 0.0001 (F - 32)] (19) 

Note. — One cubic inch of mercury at 32°F. weighs 0.4912 lb. 
The information in Table II will usually obviate the need of 
using Eq. (19). 

In case the elevation is known and no barometer available the 
problem can be solved as follows : 
Let p s = pressure of air at sea level, 
w s = weight of air at sea level, 
p x , Wx be like quantities for any other elevation. 

Then in any vertical prism of unit area and height dh we have 

dp x = w x dh. 
But 

— = — : therefore dp x = — p x dh, 
w, p s p 3 

or 

dh = — -*-^, and therefrom h = — X log — > 

W s p x W s ° p a 

where p a is the pressure at elevation h above seal level. Sub- 
stitute for w s its equivalent 

w - = s£kt and we get 53§s* = Iog f; 

Whence 

l0g e p a = loge p s - 



53.35 * 



Making p s = 14.745 and adopting to common logarithm and 
Fahrenheit temperatures, 

logio Pa = 1.16866 - J22^r+"460T (20) 

Table V is made up by formula (20). 



CHAPTER II 
MEASUREMENT OF AIR 

Art. 18. General Discussion. — Progress in the science of com- 
pressed-air production and application has evidently been hin- 
dered by a lack of accurate data as to the amount of compressed 
air produced and used. 

The custom has been almost universal of basing computations 
on, and of recording results as based on, catalog rating of compres- 
sor volumes — that is, on piston displacement. 

The evil would not be so great if all compressors had about the 
same volumetric efficiency, but it is a fact that the volumetric 
efficiency varies from 60 to 90 per cent., depending on the make, 
size, condition and speed of the machine, no wonder, then, that 
calculations often go wrong and results seem to be inconsistent. 

There are problems in compressed-air transmission and use 
for the solution of which accurate knowledge of the volume or 
weight of air passing is absolutely necessary. Chief among these 
are the determination of friction factors in air pipes and the 
efficiency of compressors, pumps, air lifts, fans, etc. 

Purchasers may be imposed upon, and no doubt often are, 
in the purchase of compressors with abnormally low volumetric 
efficiencies. Contracts for important air-compressor installation 
should set a minimum limit for the volumetric efficiency, and the 
ordinary mechanical engineer should have knowledge and means 
sufficient to test the plant when installed. 

There is little difficulty in the measurement of air. The 
calculations are a little more technical, but the apparatus is as 
simple and the work much less disagreeable than in measurements 
of water. 

At this date (1917) practice does not seem to have settled on 
a standard method of measuring quantities of air; but current 
literature shows that the subject is receiving what seems to be 
the long-delayed attention that it deserves. 

In any case where the air or gas to be measured will have a con- 
stant density and it is necessary only to get the rate of flow at any 
time, the apparatus and methods applicable would be as simple 
3 33 



34 



COMPRESSED AIR 



as those applied in measuring water, but the problem is not so 
simple when it is necessary to record the total flow (weight) dur- 
ing a considerable time during which the pressure and density 
may vary between wide limits. Though there are some appara- 
tus that the makers claim will do this, the problem does not seem 
to have been solved in a satisfactory way. 

Art. 19. Apparatus for Measuring Air. — Several methods of 
measuring the rate of flow of air at the time of observation (or 
with pressure and temperature constant), that have been pro- 
posed and tried, will be briefly noted as follows: 1 

(a) The Venturi Meter. — The principle is identical with that of 
the venturi water meter, but it is necessary to determine the 
coefficient over a range covering all pressures under which it may 
be used. This coefficient may not change with pressure, but if 
so the fact has not been ascertained. 

(b) The "Swinging Gate," 
Fig. 8a. — The air flowing in 
the direction of the arrow 
swings the gate open. The 
angle of opening depends on 
the weight of the gate, and 
on the density and velocity 
of the air. Every gate will 
have a special set of coeffi- 
cients and these would have to cover the whole field of velocities 
and densities. 

(c) The Thermal Method. — In this scheme the air is passed 
through an enlargement of the pipe in which there is placed an 
exposure of a great surface of wire, the wire being heated by a 
measured electric current. The temperature of the air is meas- 
ured before and after passing over the heated wire. The weight of 
air passing can be expressed in terms of the rise of temperature 
and the electric current absorbed. The objections are: Expen- 
sive apparatus, requiring great sensitiveness, and liability to error 
through various sources, among which is the humidity of the air. 

(d) Mechanical Meters. — This class includes common gas 
meters. They are satisfactory for commercial purposes and for 
such capacities as are covered by stock sizes. For large volumes 
they become expensive and the coefficient is always liable to 
variation, that is, the record may become inaccurate due to 

1 See Compressed Air Magazine, vol. 16, p. 6255. 




Fig. 8a. 



MEASUREMENT OF AIR 35 

corrosion or fouling of the mechanisms. Such meters show only 
the total volume that has passed between readings but unless the 
pressure and temperature are constant the record does not show 
the quantity or weight. 

As stated above, none of these methods will apply when it is 
necessary to determine the total weight passing during a pro- 
longed time in which the pressure varies. If in cases (a) and 
(&) the pressure is constant and the velocity only changes, a 
continuous recording apparatus could be attached to make a 
graph giving time and differential head in case (a) or time and 
swing of gate in case (b) from which cards the total volume could 
be integrated. If simultaneously another graph be taken show- 
ing time and pressure the two could be used to work out weights. 

If inventors could go this far, they could afford to neglect 
temperatures in commercial work. However, the cost of the 
apparatus and the labor of determining the proper coefficients 
seem to bar any of the above from general use. 

Art. 20. Measurement by Standard Orifices. — For reasons of 
economy, simplicity and accuracy, it seems that practice will 
settle on the standard orifice for determining the flow of air. 
For this reason the method and apparatus are described in 
detail. 

The standard orifice is the same as that specified for the meas- 
urement of water, that is, an orifice in a thin plate (or with sharp 
edges). In this article only circular orifices will be considered. 
These may be cut in any sheet metal up to ~% in. thick. The 
standard conditions shall be that the drop in pressure in passing 
through the orifice shall not exceed 6 in. head of water. 

With this restriction of conditions the change of temperature 
and of density of the air while passing the orifice may be neglected 
in commercial operations without appreciable error. This 
very much simplifies the formulas and reduces the chances of 
error. 

With these standards, experiments show coefficients for air 
more nearly constant than for water. 

Art. 21. Formula. Standard Orifice under Standard Condi- 
tions. — 

Let p = absolute pressure of air approaching the orifice = rp a , 
Q = weight of air passing per second, 
w = weight of a cubic foot of air at pressure p, 
d = diameter of orifice in inches, 



36 COMPRESSED AIR 

i — pressure as read on water gage in inches, 
t = absolute temperature of air (F), 
c = experimental coefficient. 

When change of temperature and of density can be neglected, 
the theoretic velocity through an orifice is 

s = \2gh 

where h is the head of air of uniform density (w) that would 

produce the pressure head i. 

Hence 

i 62.5 ,, , L i 6.25 

h = 77: > therefore s = \ 2g — - — • 

12 w v 12 w 

But Q = w X a X s where a equals the area of orifice in 

d 2 
square feet = t tttttt" Inserting these values and putting w 

under the radical, there results 



6 " 4^144 V^ 62 ^ ^ 



but 



w = 



53.35* 
therefore 

Q = 0.0136d 2 A I- rp a ' where p a ' is in pounds per square foot, 

= 0.1639d 2 A /- rp a where p a is in pounds per square inch. 

To this must be applied the experimental coefficient c so the 
formula becomes 

Q = cX 0.m9d 2 yjjrp a (21) 

For distilled water and dry air the equation would be 
Q = cX 0.1645d 2 ^ rp a . 

In very precise determinations the weight of air should be 
determined to accord with its humidity (see Art. 8a). This 
value of w would then go into Eq. (a) above. 

When working with an orifice set in a low-pressure drum, the 



MEASUREMENT OF AIR 



37 



product rp a can be most readily gotten by adding to p a the 
quantity 0.036* which is the pressure on a square inch due to 
a head i. Thus rp a = p a + 0.036*. 

If mercury be the liquid in the U-gage and barometer heights 
be inches of mercury, then 

li 



Q = c X 0.1U7d\-h 

where h = barometer height + i (i being inches of mercury). 

It will often be convenient to compute the weight of air when 
pressure is in inches of mercury. 
Then 

w a = 1.321 y (21a) 



The apparatus to be used in combination with this formula 
depends on whether the measured air is to be discharged directly 
into the free atmosphere or is to be retained in the pipe system 
under pressure. 

Art. 22. Apparatus for Measuring Air at Atmospheric Pres- 
sure. — This is the simpler of the two cases and is the one most 
easily applied in a single test of an air compressor. The essentials 
are indicated in Fig. 9. 




Fig. 9. 



A = compressed-air pipe, 
B = closed box or cylinder, 
T = throttle, 

b = baffle boards or screen, 
H = thermometer, 
C = cork, 

= orifice in thin metal plate (Standard), 
U = bent glass tube containing colored water, 
G = scale of inches. 



38 COMPRESSED AIR 

The box B may be made of any light material, wood or metal. 
The pressure will be only a few ounces and the tendency to leak 
correspondingly slight. The purpose of the throttle T is to 
control the pressure against which the compressor works. The 
appropriate orifice can be determined by a preliminary compu- 
tation, assuming i at say 3 in., or use Plate I. 

Art. 23. Coefficients for Large Orifices. — Experiments were 
made at Missouri School of Mines in 1915 to determine the co- 
efficient, c, to apply in formula (21) in case of large orifices up to 
30 in. in diameter and 30 by 30 in. square. The scheme being 
as follows: 1 

Having a fan or blower of capacity and pressure sufficient for 
the purpose, direct the discharge into a conduit across which 
place one partition containing the appropriate number of small 
standard orifices for which the coefficient is known and in an- 
other partition place the large orifice. Then the same quantity 
of air passes through the group of small orifices and the single 
large orifice, and by observing the water gage at each partition 
the relation between the coefficients can be found thus : 

Let Ci be the unknown coefficient of the large orifice, 
c 2 be the known coefficient of the small orifices, 
n be the number of small orifices open, 
d be the diameter of the small orifices, 
D be the diameter of the large orifices. 

Then by formula (21) 

Q = a X 0.1639Z) 2 A /- 1 Vl = c 2 X 0.1639wd 2 J*- 2 p 2 

Sub 1 and sub 2 indicating symbols at the large and small orifice 
partitions, respectively. 

Now it can be shown that where the drop in pressure is only 
a few inches (water gage) the factors 



may be taken as equal, especially so if the water gages be 
nearly equal at the two partitions. Hence we may express 
the relation of the two coefficients, thus 

r _ p (nd 2 fk 
1 Missouri School of Mines Bulletin, vol. 2, No. 2, November, 1915. 



MEASUREMENT OF AIR 



39 



ZT----T-----V--" 






















































































































































































il\ :;: 


p iS 


:|::::::::::::|:- 


\ : 


1 . : 
















\ 


• ; - 




\ 


\-\ '• ■ i j ! • - 
































































:-:::■ j\ 


4-LU ILL . - 


PLATE I " 

Appropriate Orifices E: 
for given- Volumes of Air:: 
Low Pressure Drum 
For High Pressure Drum :: 
Divide hv (r 1 Vi 


T^l 1 


: 














































































































































+ q * t* O O 

X _i_ co t- eo so 






















. — ^_ — j^ — | 


-lili- 


— .-OHO 


|[ 




- 


4- 




- 


. 




— 



































































































































































































































































































































































































































































































































































































































































































































































































































































































saqoui u; saoutio jo ja;eureiQ 



40 COMPRESSED AIR 

Similarly, when the large orifice is rectangular with area = a, 

nrd 2 to 



Ci — C2 1 

\ 4a \ii 

For convenience let K represent the factor in parenthesis; then 
C\ = KC%. 

In the experiments referred to, the following results were 
obtained : 

Seventy-seven 3j^-in. orifices passing to one 30-in. round K = 1.01. 

Fifty 3%-in. orifices passing to one 24-in.>ound K = 1 .00. 

Twenty-six 3^-in. orifices passing to one 18-in. round K = 0.996. 

Fifty-eight 3JHri n - orifices passing to one 18 by 30-in. rectangle. K = 1 .005. 

Sixty 3^-i n - orifices passing to one 24 by 24-in. rectangle K = 1 . 014. 

Thirty-four 33^-in. orifices passing to one 18 by 18-in. rectangle . K = . 998. 

From the above it is evident that for commercial purposes the 
coefficients for these large orifices may be taken as equal that of a 
33^-in. orifice (see Table VIII). Errors in reading water gages 
will probably exceed that made by such an assumption. 

Accepting the coefficients shown in Table VIII, those for large 
orifices are as shown in Table Villa. 

As a result of these experiments it is evident that large orifices, 
conforming to standard conditions, can be used with as much 
accuracy as in case of small ones. 

This being accepted, there is available for testing large fans 
and blowers the most reliable of all methods of measuring the 
flow of fluids, that is orifice measurement. Note that one 30-in. 
round orifice will pass about 25,000 cu. ft. per minute under 
4-in. water pressure. 

Where very large fans are to be tested several orifices can be 
set in a conduit wall. For such cases accurately constructed 
wood orifices would probably be entirely reliable and could be 
put in at moderate cost. 

Art. 23a. Notes on Water Gages. — Experience with water 
gages, and in efforts to improve on the plain water gage, while 
doing this work may be of interest. 

In such a gage (any liquid) when oscillations (not gradual 
changes of pressure) interfere with the readings, a few bird shot 
(filling the tube about an inch) will prevent oscillations and yet 
permit sufficient sensitiveness under changing pressure. 

Any coloring matter is liable to cause error by changing the 
specific gravity of the water. 



MEASUREMENT OF AIR 41 

Makers of some special gages recommend the use of gasoline 
of known specific gravity, instead of water, as it is lighter and 
therefore more sensitive. On trial it was found that if the two 
columns of the gage, above the liquid, are unequal in height, the 
presence of gasoline gas in the high column will unbalance the 
fluid columns and cause error. Often one arm of the gage is 
continued in a rubber tube. This will in effect be an extension 
of the column. In a gage in which the two columns have equal 
bore, or caliber, throughout, the sum of the two column readings 
will be constant as long as the volume of liquid in the gage does 
not change. In attempting to utilize this fact in a gage filled 
with gasoline it was found that the gasoline evaporated so fast 
as to render the scheme inapplicable. The same liability to 
inaccuracies exist in any of the combination gages in which both 
water and gasoline are used. 

Where much work is to be done while pressures are changing, 
the best scheme is to get a gage in which the sum of the readings 
is constant; use water or mercury; find the sum of the two column 
readings ^nd then read only one column. 

Let s = sum of column reading, 

h = reading of upper column of liquid, 
I = reading of lower column of liquid. 
Then i = 2 (h - y 2 s) or i = 2 (%s - I). 

Experience in this work in which thousands of readings of 
fluid pressure gages have been made under a variety of conditions 
and with a variety of gages, leads those who have done most of 
the work to the conclusion that most reliable results can be got- 
ten with pure water in a plain U-tube fastened vertically over 
a scale tacked to a plane board; the arms of the tube about 2-in. 
apart and the horizontal ruling of the scale extending under both 
arms of the gage. The readings to be taken with the assistance 
of a small draftsman's triangle held with the side resting against 
the vertical glass tube and edge against the scale, parallax being 
avoided by bringing the eye so that the upper edge of the tri- 
angle and the lines on the scale are projected parallel and both 
seen crossing the gage column as illustrated in the photograph. 
(Note that the eye of the camera was not in the correct position.) 

Art 24. Apparatus for Measuring Air Under Pressure with 
Standard Orifices. — In the ordinary case when it is desired to 
know the quantity of compressed air passing through a pipe with- 



42 



COMPRESSED AIR 



out sacrificing the pressure, the orifice drum must be made strong 
enough to withstand the high pressure and the U-gage de- 
scribed in the previous case must be replaced by a differential 
gage which must also be strong enough to withstand the pres- 
sure. The essentials are embodied in the illustration, Fig. 10, 




Fig. 9a. — Method of reading water gages. 



which also suggests a convenient scheme for attachment to an 
air main. 

The several essentials are: 

VxVzVz = valves for controlling the path of the air, 
U = unions for detaching apparatus, 
bbibi = baffles for steadying the current of air, 



MEASUREMENT OF AIR 



43 



= orifice, 

T = thermometer set through a gland, 
G = pressure gage, 
gg% = glass columns of the differential gage, 
C = cocks for convenience in manipulating the differ- 
ential gage. 
The manipulation of the apparatus Fig. 10, is as follows: 
To charge the differential gage close C\, C 4 and C 5 , open C 2 and 
C 3 and pour in the desired amount of liquid. Then close C 2 and 
C 3 and open d and C5. 

To pass the air through the measuring drum, open Vi and V3 
and close V\. 



Note: Both legs of the gage 
should be tapt into the drum close 
beside the orifice. 




Fig. 10. 

Art. 25. Coefficients and Orifice Diameters for Measurements 
at High Pressures. — Unless evidence to the contrary is shown, it 
is reasonable to assume that the same coefficients would apply to 
the orifice in the high-pressure drum, Fig. 10, that have been 
determined for the low-pressure drum, Fig. 9. However, for the 
same Q, i, t and c the diameters, d, must differ according to the 
following : 



44 COMPRESSED AIR 

Let di and p\ be the orifice diameter and air pressure respect- 
ively in the high-pressure drum, and note that the pressure in 
the low-pressure drum may be taken for this purpose as p a . 
Then 

. Qi = Q = C X 0.1QZ9d\ l - p a = cX 0.163W J.- Pl . 

Whence 

di = % (22) 

since p\/p a = r. 

By this relation the appropriate orifice can be determined from 
the curye, Plate I, by dividing the diameter ordinate by (r) Vi . 

The size drum necessary to measure a given volume of free air 
when under pressure is not as large as might be supposed before 
computations are made. For instance, with i = 3 in., T = 60°F. 
and c = 0.60, a 3-in. orifice will pass 570 cu. ft. of free air per 
minute when compressed to 100 lb. If this 3-in. orifice be placed 
in a drum 8 in. in diameter, the velocity of the compressed air 
within the drum will be 3.5 ft. per second, which is conservative. 

Example 25. — In a run with the apparatus shown in Fig. 9, the 
following were the records: d = 2.32 in., i = 4.6 in., T = 186°F. 
inside drum, T = 86°F. in free air, elevation == 1,200 ft. 

Find the weight and volume of air passing per minute. 

Solution. — From Table II interpolating for 86° in the line 
with 1,200 elevation we get w a = 0.0700 and p a — 14.1. Add to 
p a the pressure due to i ( = 0.036 X 4.6) and we get p a = 14.26. 
In Table VIII the coefficient for d = 2.32 and i = 4.6 is 0.599. 
These numbers inserted in Eq. (21) give 

Q = 0.599 X 0.1639 X (2.32) 2 ^/^ X 14.26 = 0.1684 lb. per 

second; and — — ~^ = 144.3 cu. ft. per minute of free air. 

Should there be doubt about the coefficients being the same for 
both high- and low-pressure drums, and we are willing to accept 
these now published for low-pressure drums, we can determine 
that of the high-pressure drum by placing the two drums in 
tandem, the same quantity of air passing through the high- and 
low-pressure drums in succession. Then letting sub 1 refer to 
the high-pressure drum we have the equation, 

Q = d X 0.1639di 2 J 1 - pi = c X 0.1639d 2 J~p a . 



MEASUREMENT OF AIR 45 

Whence 



tf.j/tyi-hU (23) 



d\* i hpa 
Pi 

In extensive experiments at Missouri School of Mines in 1915, 
the coefficients proved to be equal so far as practical applications 
would be concerned though the high-pressure coefficients seemed 
to be slightly less. The experiments were not conclusive. See 
description of oil differential gage, Appendix D. 

In advocating the standard-orifice method of measuring air it 
should be noted that the coefficient of an orifice is not liable to 
change with time and that the necessary apparatus can be made 
up in any reasonably well-equipped shop of a compressed-air 
plant. 

The method as presented is adapted only to show the rate of 
flow at the time of observation. To determine the quantity 
passed during any prolonged period a continuous recording 
apparatus would have to be attached that would show both the 
value of i and of p. The factor t might be assumed constant in 
most cases in practice but even then the apparatus would be 
intricate, delicate and expensive. 

It may be stated then that there are no satisfactory means now 
available to measure the quantity of air passed during a definite 
time where pressure and velocities vary. However, the obstacles 
are not insurmountable. 

Art. 26. Discharge of Air through Orifice. Considerable 
Drop in Pressure. — Referring to Figs. 9 and 10, when the differ- 
ence in pressures pi and pi is considerable, we cannot neglect the 
change of density and of temperature. 

To analyze this case we must start from the equations of energy 
at sections 1 and 2, inside and outside the orifice, the energy in 
each case being part kinetic and part potential. 

Thus 

~2g + PlVl = ~2g~ V%V2 ^ 

or 

Qsi 2 . n , Qs 2 2 . n , 
~2f + Qdi = -^ + Qch 

where c = 53.35 for 1 lb. (see Art. 2). 
Whence 

Si 2 . ■ s 2 2 , 



46 COMPRESSED AIR 

Now in any practical case the velocity of approach s x to the 

Si 2 

orifice can be made so small that the numerical value of 77- is 

2g 



so small as compared with ch 2 that it can be neglected, if desired, 

s 2 2 
without appreciable error; but not so with the quantity -^" 

Hence we may write 

s 2 2 = 2gc(ti - t 2 ). 

Substituting for t 2 its value from Eq. (12a), viz., 

we get 

ft = VW, [l - {f) V] * - VWi (l - r-f) 



Yt 



where r x is the ratio — when the escape is into free air. 



The weight passing per second is Q = w a aS 2 where a is the 
area of orifice and w a = —r in which again substitute for t 2 its 

CI 

value as above. These substitutions give 

= api yJ^S. [r,« (l - r^) »*] (24) 

This is a max. when 



r x = 



B-+.1 



When w = 1.41i Q is max. when r s = 0.526. 

When n = 1.25i Q is max. when r x = 0.555. 

Any such law as this could not have been suspected except by 
mathematical analysis, and seems contrary to what would other- 
wise have been supposed. Yet experiment seems to show that 
it is correct. 

Equation (24) is not recommended as a formula for practical 
application in measuring air. 

Art. 27. Air Measurement in Tanks. — The amount of air 
put into or taken out of a closed tank or system of tanks and 



MEASUREMENT OF AIR 47 

pipes, of known volume, can be accurately determined by 
Eq. (3), viz., 

Pa" a "a P x"a V x 

- — = - or v a = - — y 

P xV x *x Pa *x 

The process would be as follows : Determine the volumes of all 
tanks, pipes, etc., to be included in the closed system, open all 
to free air and observe the free-air temperature; then switch 
the delivery from the compressor into the closed system; count 
the strokes of the compressor until the pressure is as high as 
desired; then shut off the closed tank and note pressure and tem- 
peratures of each separate part of the volume. Then the formula 
above will give the volume of free air which compressed and 
heated would occupy the tanks. From this subtract the volume 
of free air originally in the tanks ; the remainder will be what the 
compressor has delivered into the system. Note that the com- 
pressor should be running hot and at normal speed and pressure 
when the test is made for its volumetric efficiency. 

Usually the temperature changes will be considerable, but if 
the system is tight, time can be given for the temperature to 
come back to that of the atmosphere, thus saving the necessity 
of any temperature observations. 

Where a convenient closed-tank system is available, this 
method is recommended. 

This method— that is, Eq. (3) as stated above — was used to 
determine the quantity of air passing the orifices in the experi- 
ments by which the coefficients were determined as given in 
Art. 21, Table VIII. 

The varying volumetric efficiencies with changes of tempera- 
ture and pressures can be shown very impressively by starting 
with compressor cool and the air in tanks at atmospheric pres- 
sure. Then note the number of revolutions that bring the pres- 
sure up to say 20, 40, 60, 80 lb., and so get the data for volumetric 
efficiencies in each interval. In the first it may be found as high 
as 95 per cent, while in the last interval it may fall below 60 per 
cent, in small compressors. Of course, that in the last interval 
is that by which the compressor should be judged. 

Example 27. — A tank system consists of one receiver 3 ft. in 
diameter by 12 ft., one air pipe 6 in. by 40 ft., one 4 in. by 4,000 
ft. and a second receiver at end of pipe 2 ft. in diameter by 8 ft. 
A compressor 12 by 18 in. with 1^-in. piston rod puts the air 



48 COMPRESSED AIR 

from 1,250 revolutions into the system, after which the pressure 
is 80-gage and temperature in first receiver 200°, while in other 
parts of the tank system it is 60°. Temperature of outside air 
being 50°, p a = 14.5 per square inch. Find volumetric efficiency 
of the compressor. 

Solutions. — Volumes (from Table X) : 

First receiver 84.84 cu. ft. 

6-in. pipe 7.841 

4-in pipe 349.20 382.16 

Second receiver. ..... 25. 12 J 

Total 467.00 in tank system. 

Piston displacement in one revolution = 2.338 cu. ft. (piston 
rod deducted). 

(T) t \ V 

— ^1 X — note that the quantity in paren- 
Pa ' lx 

thesis is constant and therefore a slide rule can be conveniently 
used, otherwise work by logarithms 

. _ . (80 + 14.5) (460 + 50) v , 84.84 , nf70 

v a in first receiver = - ^-^ — X 46Q + 2QQ = 417.2 

v a in 6-in. pipe, 4-in. pipe and second receiver with total 

volume 382.16 and t = 60° = 2,447. 1 

Total 2,864.3 

Original volume of free air , 467 . 

Volume of free air added 2,397.3 

2,397.3 + 2.338 = 1,028. 
Therefore the volumetric- efficiency is 

E = 1,028 -T- 1,250 = 82 per cent. 



CHAPTER III 
FRICTION IN AIR PIPES 

Art. 28. — In the literature on compressed air many formulas 
can be found that are intended to give the friction in air pipes 
in some form. Some of these formulas are, by evidence on their 
face, unreliable, as for instance when no density factor appears; 
the origin of others cannot be traced and others are in incon- 
venient form. Tables claiming to give friction loss in air pipes 
are conflicting, and reliable experimental data relating to the 
subject are quite limited. 

In this chapter are presented the derivation of rational for- 
mulas for friction in air pipes with full exposition of the assump- 
tions on which they are based. The coefficients were gotten 
from the data collected in Appendix B. 

Art. 29. The Formula for Practice. — The first investigation will 
be based on the assumption that volume, density and tempera- 
ture remain constant throughout the pipe. 

Evidently these assumptions are never correct; for any de- 
crease in pressure is accompanied by a corresponding increase 
in volume even if temperature is constant. (The assumption of 
constant temperature is always permissible.) However, it is 
believed that the error involved in these assumptions will be 
less than other unavoidable inaccuracies involved in such 
computations. 

Let / = lost pressure in pounds per square inch, 
I = length of pipe in feet, 
d = diameter of pipe in inches, 
s = velocity of air in pipe in feet per second, 
r = ratio of compression in atmospheres, 
c = an empirical coefficient including all constants. 

Experiments have proved that fluid friction varies very nearly 

with the square of the velocity and directly with the density. 

Hence if k be the force in pounds necessary to force atmospheric 

air (r = 1) over 1 sq. ft. of surface at a velocity of 1 ft. per 

4 49 



50 COMPRESSED AIR 

second, then at any other velocity and ratio of compression the 
force will be 

F x = ks 2 r, 

and the force necessary to force the air over the whole interior 
of a pipe will be 

F = ^IX krs\ 

and the work done per second, being force multiplied by distance, 
is 

Work = — r X krs 3 . 
12 

Now if the pressure at entrance to the pipe is / lb. per square 
inch greater than at the other end, the work per second due to 
this difference (neglecting work of expansion in air) is 

Work = f-±- s. 

Equating these two expressions for work there results 

.ird 2 ivd 77 , 

/ -j- s = ^2 lkrs ' 

or 

f-~A rsi (25) 

Now the volume of compressed air, v, passing through the pipe 
is, in cubic feet, 

7T^ 2 

V ~ 4 X 144 S 

and the volume of free air v a is rv. 
Therefore 

Va = 4X1U X rS 
and 

2 = (4 X 144) V 
S " " 7r 2 d 4 r 2 

Insert this value of s 2 in Eq. (25) and reduce and the results 
4 _ /4 X 144\ 2 1 v a 2 



'=12*( 



7 & 



or 

I v a * 



/ = c-h — (26) 

J d 5 r 



FRICTION IN AIR PIPES 51 

where c is the experimental coefficient and includes all constants. 
From Eq. (26), 

/clv a 2 \ ** 
*-(-£■) (27) 

From the data recorded in the appendix the coefficients for 
formula (26) were worked out, first using the actual measured 
diameters, second using the nominal diameters. The average 
of the coefficients for each size pipe were then platted and the 
results tabulated as shown on Plate II. In studying this plate 
it should be borne in mind that the vertical scale is ten times that 
of the horizontal which exaggerates the irregularities of the 
coefficient. 

These studies reveal conclusively that c is practically independ- 
ent of r and of s (the velocity in pipes), and that it increases as 
the diameter decreases. If temperature has any effect, it could 
not be detected. Since the friction varies inversely as the fifth 
power of the diameter, it is very sensitive to any variation in 
the diameter. Hence, if the greatest possible accuracy is de- 
sired, the computations should be based on the measured diame- 
ter and the coefficient taken from the curve AB, Plate II. 
If the actual diameter is unknown and the computer must use 
nominal diameters, the coefficient should be taken from the line 
CD. In any case computations of friction loss in commercial 
pipes of less than 1 in. in diameter will be unreliable on account 
of the relative great effect caused by small obstructions and 
irregular diameters. 

Table IX is computed from Eq. (26) and is self-explanatory. 
It affords a direct and easy determination of friction losses in 
air pipes. 

A further study of the coefficients found by the curve AB, 
Plate II, shows that the logarithms of c and d plat to a straight 
line from which is obtained the relation 

= 0.1025 
C d°- sl ' 



This inserted in Eq. (26) gives 

10 
rd 



0.1025lv a > 



or 



, _ 0.1025 IvJ . 

J ~ 3,600 rd 5 - 31 {Zm) 



52 



COMPRESSED AIR 



c 

6 
o 
O 

a 

5 










- 
















»TTTr 








„ 




_ 






. 




*, 












































. Oj 








j 




a 


uo Suia't; 


OOOOoO OOOOoOO 
















































" 




























































j 










-i 






OOOOOOOO 1 


01 
O 


























































- 


















































' 










































































as 


9V eAJt»o 
no SuiX^j 




































































.110 
.100 
.089 
.082 
.072 
.0G6 
.032 
.059 
.056 
.053 
.050 
.043 














































1 




























































/ 












/ 




























































/ 




pa^ndtaoQ sy 


rM 1— 1 O OOO OOO OO 












































































































































































































































i 






















































f 












/ 






pajtisBajv: 


0.83 

0.82 

1.07 

1.63 

2.07 

2.874 

3.937 

4.921 

5.960 


n 














































/ 












1 
















































/ 
















































































" 






























































JBECUOJJ 


^ ro\ 1 " H ^' M^-^^sst-O'^^ 


















































































































1 


































































































































































































































































1 






























































































































































































































































j 




































































































































/ 




































































































































/ 








, 




























































































































/ 
1 
































































































































































































































































PLATE II 
„ 1 v 
Coefficients c for Formula f=c -p-jr 

O Indicates Average C Computed fron 

Measured Diameters 
A Indicates the C Adapted to Nomina 

Diameters -Computed from Values on 

Curve AB 




















































































































































































1 








/ 




























































































/ 




































































































/ 














































































































































































































/ 




































































































/ 




























































































/ 






/ 






















































































/ 






/ 






























































































/ 

1 














































































































































































































1 
































































































































































































1 




































































































/ 






/ 






























































































/ 
1 












































































































































































































, 


































































































































































































j 






' 






























































































/ 




/ 
































































































/ 




/ 






































































































































































































/ 
































































































, 




































































































/ 


I 


































































































/ 


1 


































































































// 






































































































































































































! 




































































































/ 


































































































/ 


/ 




































































































/ 
































































































/ 




/ 


































































































































































































































































/ 








































































































































I 




































































































































/ 




































































































































/ 




































































































































/ 










































































































































































































































































































































































































, 




































































































































/ 




































































































































/ 














































































































































































































































































































































































































































































































































, 




































































































































/ 




































































































































/ 




































































































































1 






































































































































































































































































































































































> 




































j 




























































































s 








































/ 




































































































































/ 




































































































































' 




































































































































1 


































































































































' 










































_< 




































































































































d 
























































































































































































































/ 





















































































































































































































































































































































































































































































































































































































































































































































































































































(1 0„ }USPUP°D 



FRICTION IN AIR PIPES 



Chart for Solving Formula /= -1025 * _ 



7-d 6 - 31 x3G00' 
/= Friction Loss in Pounds per Square Inch. 
I = Length, of Pipe in Feet. 
•y= Cubic Feet of Free Air per Minute. 
*"= Eatio of Compression 
d= Diameter of Pipe in Inches. 

The Dependent Factors {fr), v and d Lie in 
a Straight Lino. To get the Friction Loss in 
1000 Feet; Divide the (fr) by r. 

Friction of Gasses will be Proportional 
the their Specific Gravities. 



-200 
-190 
-ISO 
-170 
-ISO 
-150 
-140 
-130 
-120 

-no 

-100 



or v"' = 35.13 (fr) d 



80,000 
7.-..HU0 
70,000 
65,000 
60,000' 
55,000 
50,000 
45,000 
40,000 
35,000 
30,000 
28,000 
20,000 
24,000 
22 000 
20,000 
18,000 
16,000 
14,000 
12,000 
10,000 
9000 
8000 
7000- 
6000- 

5000. 

4500- 
4000- 
3500- 
3000- 

2500- 

2000. 
1800- 

1600- 
1400- 
1200- 

1000- 
900- 
800- 
700- 
600- 

500- 
450- 
400- 
350- 
300- 

250- 

200- 



53 

12 1 



10 



6- 



4V 2 - 



ZV 2 - 



2V 2 ~ 



2- 



70- 
60- 



1 3 4- 



1H- 



1V 4 - 



t 1 



Plate III. 



1- 



54 COMPRESSED AIR 

where v a is in cubic feet per minute. Log ' fif)f) = 5.4544. 

This equation gives results practically indentical with those 
from Eq. (26) when c is taken from the curve AB. It is almost 
as easy of solution and has the advantage that it is independent 
of a table of coefficients. 

Plate III is a logarithmic chart for solving Eq. (28a). 

Since such a chart can handle only three variables, the product 
fr is taken as a single variable and I as 1,000 ft. 

To solve the equation by this chart, lay a straight edge (or 
stretch a thread) over the chart. The three numbers under 
the line will satisfy Eq. (28a). 

Example 28a. — What pressure will be lost in a 4-in. pipe 
5,000 ft. long when transmitting 1,200 cu. ft. of free air per 
minute compressed to 7 atmospheres (r = 7). 

A thread stretched over 4 in. and 1,200 cu. ft. crosses the fr 
line at 25, then 25 4- 7 = 3.6 and 3.6 X 5 = 18 lb. 

Since the process of designing such charts as Plate III has not 
appeared in any of the well-known text-books, the author has 
made it available in Appendix B. 

The following formula is that derived by Church for loss by 
friction in air pipes : 

2 _ 4clQ 2 p 2 



Vr - Vi 



gdA^Wz 



In this pi and pi are pressures at points on the pipe distance I 
apart, p\ being the less pressure, A is the area of the pipe and c 
some experimental coefficient. The other symbols are as used 
elsewhere in this article. 

Frank Richards recommends a simplification of Church's 
formula by assuming c constant and a temperature about 60°F. 
His formula is 

V a H 



V% - P\ L = 



2,000d E 



In the experiments at the Missouri School of Mines in 1911 
(described in Appendix C) effort was made to find the laws of 
resistance to flow of air through various pipe fittings. Facili- 
ties were not available for sizes above 2 in. in diameter and for the 
smaller sizes the results were erratic, doubtless due to the rela- 
tively greater effect of obstructions and variations in diameter 



FRICTION IN AIR PIPES 



55 



in the small pipes. The results are given below. Further re- 
search is needed along this line. 

Lengths op Pipe in Feet that GrvE Resistance Equal that 
op a Single Fitting 



Diameter 
of pipe, 
inches 


Elbows 90° 


Unreamed 
joints, 2 ends 


Reamed joints 


Return binds 


Globe valves 


y* 


10.0 


2-4 


7 


10.0 


20 


% 


7.0 


2-4 


7 


7.0 


25 


i 


5.0 


2-4 


7 


5.0 


40 


m 


4.0 


2-4 


7 


4.0 


45 


2 


3.5 


2-4 


7 


3.5 


47 



Tests on resistance in 50-ft. lengths of rubber-lined armored 
hose, with their end fittings such as is used to connect with com- 
pressed-air tools, were made with average result as follows : 



Diameter of hose, inches 


H 


1 


IK 




Resistance in 50-ft. length 


20^ 
r 


4.5 ~ 
r 


V a 2 

2.6 — 
r 



Finally it is important to note that in cases where gases other 
than air are under consideration the friction losses will be di- 
rectly proportional to the specific gravity of the gas, for instance 
if the gas has a specific gravity of 0.8 the friction will be 0.8 of 
that for air under the same conditions. 

The rate of flow of air or gas through a long pipe of uniform 
diameter can be computed approximately by observing / for 
distance I; then 

in case of air, or 



cl 



Va = 



frd 5 



c X 0.81 



in case of gas of 0.8 specific gravity. 

This formula may be of value in determining the flow of natu- 
ral gas through long pipes. 

It may be well to note here that the deposit of solid matter 



56 COMPRESSED AIR 

(paraffines and asphalts) out of natural gas may seriously obstruct 
the pipes and render such computations altogether inaccurate. 

Example. — 1,600 cu. ft. per minute of free air is supplied to a 
mine at a pressure of 7 atmospheres (r = 7) through a 4-in. 
main. At a distance of 2,840 ft. from the compressor is a 2-in. 
branch placed to take air to two 2}4 drills, requiring 100 cu. ft. 
each of free air per minute. The 2-in. pipe is 1,260 ft. long and 
has in that length two globe valves, four elbows, and eighteen 
unreamed (extra) joints. 

Each drill takes its air through 50 ft. of 1-in. hose. 

What will be the loss of pressure at the drills? 

Solution. — By formula (28a) : 

Loss in the 4-in. main: 





5.4544 


log 7 = 0.8457 


log 2,840 


= 3.4533 


5.31 log 4 = 3.1972 


2 log 1,600 


= 6.4082 
5.3159 
4.0423 


4.0423 


log .A 


= 1.2736 


.*. fi = 18.8 for 4-in. p 



Loss in 2-in. pipe. Note that the r is about 5.75 in the 2-in pipe : 

Effective length, straight 1,260 ft. 

Effective length, 2 glove valves @ 47 94 ft. 

Effective length, 4 elbows @ 3.5 14 ft. 

Effective length, 18 unreamed joints 3.0 54 ft. 

Total 1,422 ft. 

5.4544 log 5.75 = 0.7597 

log 1,422 = 3.1529 (5.31) log 2 = 1.5983 

2 log 200 = 4.6020 2.3580 

3.2093 

2.3580 
log/a = 0.8513 .'. / 2 - 7.10 for 2-in. pipe. 



v 
Loss in 50 ft. of 1-in. hose delivering 100 cu. ft., / = 4.5 — - • 

Note that the r in the hose is (after deducting the accumulated 
friction in the 4-in. and the 2-in. pipes) about 5.25. 



FRICTION IN AIR PIPES 57 

log 4.5 = . 6532 log 5.25 = . 7202 

2 log 100 = 4.0000 log 3,600 = 3.5563 

4.6530 4.2763 

4.2765 
log/ = 0.3765 .*. 2.4 for the 50-ft. hose. 

Total loss of pressure = 18.8 + 7.1 + 2.4 = 28.3. 

Evidently such computations as this should not be accepted as 
giving precise results. Such matters as the varying r, varying 
density of air as effected by temperature and free air pressure, 
irregular qualities and changing conditions of the pipes, leaks, 
and irregular demands for air all more or less effect the resulting 
loss. Nevertheless such computations are the proper guides for 
the designer. 

Art. 30. Theoretically Correct Friction Formula. — The theo- 
retically correct formula for friction in air pipes must involve the 
work done in expansion while the pressure is dropping. 

Let pi and p 2 be the absolute pressures at entrance and dis- 
charge of the pipe respectively and let Q be the total weight of 
air passing per second. 

Then the total energy in the air at entrance is 

and at discharge the energy is 

1 P2 , Qsi 2 

p-^^ + w 

The difference in these two values must have been absorbed in 
friction in the pipe. Hence, using the expression for work done 
in friction that was derived in Art. 29, we get 



= VaVa (log ^ - log ^) - jr- (s 2 2 - Si 2 ). 



^W- 



Numerical computations will show the last term, viz., 

is relatively so small that it can be neglected in any case in 
practice without appreciable error. Hence, by a simple reduc- 
tion we get 

, pi irk v , dlrs 3 , , ird 2 



58 COMPRESSED AIR 



which when substituted gives 



. pi 4 X 144fc I 2 

log e — = -7T X -j S 2 , 

p 2 I2p a d 

or considering p a as constant, 

I ' ■ Pi Z 2 

P2 d 

or 

logio P2 = logio Pi -c x ^s 2 (29) 

In Eq. (29) C\ is the experimental coefficient and includes all 
constants, s is the velocity in the air pipe and varies slightly 
increasing as the pressure drops. All efforts so far have failed 
to get a formula in satisfactory shape that makes allowance for 
the variation in s. 

In working out Ci from experimental data s should be the mean 
between the s x and s 2 , and when using the formula the s may be 
taken as about 5 per cent, greater than Si. 

Note that in the solution of Eq. (29) common logarithms should 
be used for convenience, allowing the modulus, 2.3+, to go 
into the constant C\. 

The working formula may be put in a different and possibly a 
more convenient form, thus. In the expression 

i Pi t& . . dl 
l0g e — = rrr X rs 3 

p 2 12 p a V a 

substitute for s its value 

= 4 X 144^ a 
ird 2 r 
and reduce and we get 

ha 2 

p a d 5 r 



log p 2 = log pi - c 2 — j££ (30) 



Still another form is gotten thus. The whole weight of air 
passing is v a X w a = Q, and by Eq. (13) 



Q = v a rooci and therefore v a = 



53.351Q 



53.35f u p a 

Also 

r x — ^— and w a = 



p a 53.35^ 

Substitute these in (30) and it reduces to 



FRICTION IN AIR PIPES 59 



log p 2 = log p'i - c 2 —^r b (—) (31) 

w a d b \pj 

■ctice — 

done Eq. (31) becomes 



In ordinary parctice — may be taken as constant. If this be 

w a 



log p 2 = log Vi ~ c s j h [—) (31a) 



d 5 \p x 

If f a = 525 and w a = 0.075, then c 3 = 7,000 c 2 . 

In (31) and (31a) p s varies between pi and p 2 . Careful com- 
putations by sections of a long pipe show p x to vary as ordinates 
to a straight line. Modifying the formulas to allow for this 
variation renders them unmanageable. In working out the 
coefficient p x may be taken as a mean between pi and p 2 , and in 
using the formula p may be taken as p\ less half of the assumed 
loss of pressure. 

As before suggested, common logarithms should be used in 
all the equations of this article. 

A study of the data collected in Appendix B gives values for 
c 2 Eq. (31), that, for pipes 3 to 12 in. in diameter, conform closely 
to the expression. 

c 2 = 0.0124 - 0.0004d, 

which gives the following: 

d" = 3 4 5 6 8 10 12 

C 2 = 0.0112 0.0108 0.0104 0.0100 0.0092 0.0084 0.0080 
C 3 = 78.4 75.6 72.8 70.0 64.4 58.8 56.0 

With these coefficients p x in Eqs. (31) and (31a) is to be taken 
in pounds per square inch. 

Equations (31) and (31a) are theoretically more correct than 
Eq. (26) and the coefficients of the former will not vary so much 
as those for the latter, but when the coefficients are correctly 
determined for Eq. (26) it is much easier to compute and can be 
adapted to tabulation, while Eq. (31) cannot be tabulated in any 
simple way. 

Finally it should be said that extreme refinement in computing 
friction- in air pipes is a waste of labor, for there are too many 
variables in practical conditions to warrant much effort at 
precision. 

Example 24a. — Apply formulas (26) and (31) to find the pres- 
sure lost in 1,000 ft. of 4-in. pipe when transmitting 1,200 cu. ft. 



60 COMPRESSED AIR 

free air per minute compressed to 150 gage when atmospheric 
conditions are p a = 14.0, w a = 0.073 and t a = 540. 

Solution by Eq. (20).— r = 15 °^ U = 11.71. By Table IX 

divide 23.44 by 11.71 and the result, 2 lb., is the pressure lost 
per 1,000 ft. 

Solution of Eq. (31). — The coefficient for a 4-in. pipe is 0.0108, 
and log pi = log (150 -f- 14) = 2.214844. 
Then 

1 nniAOAA nn-.no 54 w 1,000 /1,200 w 0.073\ * 

log p 2 = 2.214844 - 0.0108 ^X-^ (-% X w ) • 

The log of the last term is 3.791193 and its corresponding number 
is 0.006183. 

2.214844 - 0.006183 = 2.208661 = log p 2 . 
Whence 

p 2 = 161.7 + and pi — p 2 = 2.3. 

Art. 31. Efficiency of Power Transmission by Compressed 
Air. — In the study of propositions to transmit power by piping 
compressed air, persons unfamiliar with the technicalities of 
compressed air are apt to make the error of assuming that the 
loss of power is proportional to the loss of pressure, as is the case 
in transmitting power by piping water. Following is the 
mathematical analysis of the problem : 

pi = absolute air pressure at entrance to transmission pipe, 
p 2 = absolute air pressure at end of transmission pipe, 
vi = volume of compressed air entering pipe at pressure pi, 
02 = volume of compressed air discharged from pipe at 
pressure p2. 

Then crediting the air with all the energy it can develop in 

Pi 

isothermal expansion, the energy at entrance is piVi log — = 

Pa 

PiVi log ri, and at discharge the energy is p 2 V2 log — = p 2 02 log r 2 . 

Pa 

Hence 

efficiency E = W l °& r * = fe (32) 

P1V1 log e r x log ri 

Common logs may be used since the modulus cancels. The 
varying efficiencies are illustrated by the following tables: 



FRICTION IN AIR PIPES 61 

p a = 14.5. pi = 87. n = 6. log n = 0.7781. 



7>2 


85 
5.86 
0.7679 
0.987 


80 
5.52 
0.7419 
0.953 


75 
5.17 
0.7135 
0.917 


70 
4.83 
0.6839 
0.879 


65 
4.48 
0.6513 
0.837 


60 


7*2 


4.14 


log T2 


0.6170 


E 


0.793 









p a = 14.5. pi = 145. 


r x = 10. 


log r 1 = 


= 1.000. 




■p 2 


140 
9.66 
0.9850 
0.98 


135 
9.31 
0.9689 
0.97 


130 

8.97 

0.9528 

0.95 


125 
8.62 
0.9355 
0.93 


120 


T2 


8.28 


log T% 


0.9185 


E 


0.92 







The above examples illustrate the advantage in transmitting 
at high pressure. Of course the work cannot be fully recovered 
in either case without expanding down to atmospheric pressure, 
and to do this in practice heating would be necessary. It should 
be understood also that by reheating this efficiency can be exceeded. 

It should be noted also that the above does not apply in cases 
where the air is applied without expansion. In such cases the 
efficiency of transmission alone would be 



E = 



(Pi — Pa) V2 = n 0*2 - 1) 
(Pi — Pa) V l r 2 0*1 — 1) 



Example 31a. — What diameter of pipe will transmit 5,000 
cu. ft. of free air per minute compressed to 100 lb. gage, with a 
loss of 10 per cent, of its energy, in 2,500 ft. of pipe, assuming 
p a = 14.0? 



Solution. — r\ 



114 
14 



= 8.15; then by Eq. (30) 



log r 2 
log 8.15 



90 
100' 



Whence log r 2 = 0.8200; r 2 = 6.6, and 6.6 X 14 = 92.4. 

/ = 114 — 92.4 = 21.6 = loss of pressure. 
By Eq. (27), 

log d=\ [log (0.06 X 2,500) X (^~) *- log 

(21.6X?^±M)] 

= 0.7602, whence d = 5.75 in. 

21 6 
Otherwise go into Table IX with loss for 1,000 ft. = -^j = 8.64, 

and 8.64 X r = 8.64 X 7.37 = 63 (7.37 being the mean r). 



62 COMPRESSED AIR 

Then opposite 5,000 in the first column find nearest value to 63, 
which is 55 in the 6-in. column; showing the required pipe to be 
a little less than 6 in. 

Otherwise over Plate III stretch a thread passing over 63 on 
the fr line and 5,000 on the V a line. It will cut the d line at 
5M. 



CHAPTER IV 
OTHER AIR COMPRESSORS 

Art. 32. Hydraulic Air Compressors. — Displacement Type. — 

Compressors of this type are of limited capacity and low effi- 
ciency, as will be shown. They are therefore of little practical 
importance. However, since they are frequently the subject 
of patents and special forms are on the market, their limitations 
are here shown for the benefit of those who may be interested. 

Omitting all reference to the special mechanisms by which the 
valves are operated, the scheme for such compressors is to admit 
water under pressure into a tank in which air has been trapped 
by the valve mechanisms. The entering water brings the air 
to a pressure equal to that of the water; after which the air is 
discharged to the receiver, or point of use. When the air is all 
out the tank is full of water, at which time the water discharge 
valves open, allowing the water to escape and free air to enter the 
tank again, after which the operation is repeated. Usually these 
operations are automatic. The efficiency of such compression is 
limited by the following conditions. 

Let P = pressure of water above atmosphere, or ordinary gage 
pressure, 
V — volume of the tank. 

Then P + p a = absolute pressure of air when compressed. 
The energy represented by one tank full of water is PV and by 
one tank full of free air when compressed to P + p a is 

PaV 10g e = p a V log e T. 

Pa 

Therefore the limit of the efficiency is 

PaV lOge r p a log e T 



E 



PV 



But P — pi — p a , where p± is the absolute pressure of the com- 
pressed air. Inserting this and dividing by p a the expression 
becomes 

E = log e r __ logio r X 2.3 

r — 1 r — 1 - ' 

63 



64 



COMPRESSED AIR 



Table VII is made up from formula (33) . 

The practical necessity of low velocities for the water entering 
and leaving the tanks renders the capacity of such compressors 
low in addition to their low efficiency. 

Should the problem arise of designing a large compressor of 
this class an interesting problem would involve the time of filling 
and emptying the tank under the varying pressure and head. 
Since it is not likely to arise space is not given it. 

It is possible to increase the efficiency of this style of com- 
pressor by carrying air into the tank 
with the water by induced current or 
Sprengle pump action — a well-known 
principle in hydraulics. At the begin- 
ning of the action water is entering the 
tank under full head with no resistance, 
and certainly additional air could be 
taken in with the water. 

Art. 33. Hydraulic Air Compressors. — 
Entanglement Type. — A few compressors 
of this type have been built compara- 
tively recently and have proven remark- 
ably successful as regards efficiency and 
economy of operation, but they are 
limited to localities where a waterfall is 
available, and the first cost of installa- 
tion is high. 

The principle involved is simply the 
reverse of the air-lift pump, and what 
theory can be applied will be found in 
Art. 39 on air-lift pumps. 

Figure 11 illustrates the elements of 

a hydraulic air compressor, h is the 

effective waterfall. 

H is the water head producing the pressure in the compressed air. 

t is a steel tube down which the water flows. 

S is a shaft in the rock to contain the tube t and allow the water 

to return. 
R is an air-tight hood or dome, either of metal or of natural rock, 

in which the air has time to separate from the water. 
A is the air pipe conducting the compressed air to point of use. 




OTHER AIR COMPRESSORS 65 

6 is a number of small tubes open at top and terminating in a 
throat or contraction, in the tube t. 

By a well-known hydraulic principle, when water flows freely 
down the tube t there will occur suction in the contraction. 
This draws air in through the tubes 6, which air becomes en- 
tangled in the passing water in a myriad of small bubbles; these 
are swept down with the current and finally liberated under the 
dome R, whence the air pipe A conducts it away as compressed 
air. 

The variables involved practically defy algebraic manipula- 
tion, so that clear comprehension of the principles involved must 
be the guide to the proportions. 

Attention to the following facts is essential to an intelligent 
design of such a compressor. 

1. Air must be admitted freely — all that the water can entangle. 

2. The bubbles must be as small as possible. 

3. The velocity of the descending water in the tube t should 
be eight or ten times as great as the relative ascending velocity 
of the bubble. 

The ascending velocity of the bubble relative to the water 
increases with the volume of the bubble, and therefore varies 
throughout the length of the tube, the volume of any one bubble 
being smaller at the bottom of the tube than at the top. For 
this reason it would be consistent to make the lower end of the 
tube t smaller than the top. 

Efficiencies as high as 80 per cent, are claimed for some of 
these compressors, which is a result hardly to have been expected. 

The great advantage of this method of air compression lies 
in its low cost of operation and in its continuity. Mechanical 
compressors operated by the water power could be built for less 
first cost and probably with as high efficiency, but cost of opera- 
tion would be much higher. 

Evidently there is a limit to the amount of air that can be 
taken down and compressed by this hydraulic air compressor. 
By the laws of conservation of energy we know that the energy 
in the compressed air as expressed by formula pv log e r cannot 
exceed that of the waterfall which is Wh where W is the weight 
of water passing, or in general 

. ■ Wh 

PaV a lOge T < Wfl Or V a < : * 

p a log e r 
The limitation can also be seen from the following considerations: 



66 COMPRESSED AIR 

Let V represent the total volume of air in the whole length of 

the downcast pipe t and let A represent the area of that pipe. 

V 
Then when -r = h the downflow of water will cease, for the static 

pressure inside and outside the pipe will be equal — in this state- 
ment friction and velocity head in the pipe are neglected. A 
more correct statement would be that in order to be operative 

V S 2 

where / is the head lost in friction and s the velocity in the 
downcast. 

Evidently in this, V is the dominant number and it can be 
controlled by opening or closing some of the inlet tubes at b. 
It is by such manipulation that the most efficient working can 
be secured. 

Art. 34. Centrifugal and Turbo Air Compressors. — With the 
development of the steam turbine it has become practicable to 
deliver air at several atmospheres pressure by means of centrif- 
ugal machines. 

The very high speed at which such machines are run (up to 
4,000 r.p.m.) calls for the most perfect possible material and 
workmanship. Yet they are relatively simple, occupy small 
space, are of low first cost and are quite efficient, as compared 
with reciprocating machines to do equal service. These quali- 
ties assure this class of machine (which includes the "turbo air 
compressors") a popularity where large volumes of air are re- 
quired at a moderate and constant pressure. 

One very effective application of turbo air compressors is as 
a "booster" to large reciprocating machines, the scheme being 
to use the exhaust steam from the engines to run the steam tur- 
bines that actuate the turbo compressors. The air from the 
turbo compressors is delivered into the intake of the reciprocating 
machines. A relatively small increase in the intake pressure 
will materially influence the capacity and economy of opera- 
tion of the reciprocating machines. For example : Assume that 
the turbo machines deliver air at }?£ atmosphere, gage pressure; 
that is r = 1)^. Then if the air be cooled to its original tem- 
perature before entering the reciprocating machine, the weight 
of air handled will be increased one-half. Now assume the re- 
ciprocating machine to have been designed to compress free air 



OTHER AIR COMPRESSORS 67 

to a ratio r = 6 or about 75 lb. gage; then with the booster at- 
tached, and maintaining the same ratio (6) of compression 
within the compressor, the delivery ratio relative to atmosphere 
will be 9 or a gage pressure about 120 lb. This would be accom- 
plished without compounding and without development of any- 
more heat than without the booster. However, more work 
would be required of the reciprocating engines. Hence, in 
studying such an improvement the designer should determine 
whether the engines can meet the demand for increased power. 
The volume of air delivered by and the efficiency of centrifugal 
and turbo compressors, fans and blowers are matters understood 
by but few, seldom known, and often far from what is assumed or 
claimed. The theory underlying these subjects is somewhat 
difficult and is deferred to Chapters VIII and IX. 



CHAPTER V 
SPECIAL APPLICATIONS OF COMPRESSED AIR 

In this chapter attention is given only to those applications of 
compressed air that involve technicalities — with which the de- 
signer or user may not be familiar, or by the discussion of which 
misuse of compressed air may be discouraged and a proper use 
encouraged. 




Engine 



Fig. 12. 



Art. 35. The Return-air System. — In the effort to economize in 
the use of compressed air by working it expansively in a cylinder 
the designer meets two difficulties: first, the machine is much 
enlarged when proportioned for expansion; second, it is consider- 
ably more complicated; and third, unless reheating is applied the 
expansion is limited by danger of freezing. 

To avoid these difficulties it has been proposed to use the air at 

68 



SPECIAL APPLICATIONS OF COMPRESSED AIR 69 

a high initial pressure, apply it in the engine without expansion, 
and exhaust it into a pipe, returning it to the intake of the com- 
pressor with say half of its initial pressure remaining. The 
diagram, Fig. 12, will assist in comprehending the system. 

To illustrate the operation and theoretic advantages of the 
system assume the compressor to discharge air at 200 lb. pressure 
and receive it back at 100 lb. Then the ratio of compression is 
only 2 and yet the effective pressure in the engine is 100 lb. 

Evidently then with a ratio of compression and expansion of 
only 2 the trouble and loss due to heating are practically removed ; 
and the efficiency in the engine even without a cutoff would be, 
by Eq. (18) 72 per cent. By the above discussion the advantages 
of the system are apparent, and where a compressor is to run a 
single machine, as for instance a pump, the advantage of this 
return-air system will surely outweigh the disadvantage of two 
pipes and the high pressure, but where one compressor installa- 
tion is to serve various purposes such as rock drills, pumps, ma- 
chine shops, etc., the system cannot be applied. There should 
be a receiver on each air pipe near the compressor. 

Art. 36. The Return-air Pumping System. — Following the 
preceding article it is appropriate to describe the return-air 
pumping system. The economic principle involved is different 
from that of the return-air system just explained. 

The scheme is illustrated in Fig. 13. It consists of two tanks 
near the source of water supply. Each tank is connected with the 
compressor by a single air pipe, but the air pipes pass through a 
switch whereby the connection with the discharge and intake 
of the compressor can be reversed, as is apparent on the diagram. 
In operation, the compressor discharges air into one tank, thereby 
forcing the water out while it is exhausting the air from the 
other tanks, thereby drawing the water in. The charge of air 
will adjust itself so that when one tank is emptied the other will 
be filled, at which time the switch will automatically reverse the 
operation. 

The economic advantage of the system lies in the fact that the 
expansive energy in the air is not lost as in the ordinary dis- 
placement pump (Art. 37). The compressor takes in air at vary- 
ing degrees of compression while it is exhausting the tank. 

The mathematical theory, and derivation of formulas for 
proportioning this style of pump are quite complicated but 
interesting. 



70 



COMPRESSED AIR 



Preliminary to a mathematical study for proportioning the 
installation it is well to follow a cycle in its operation: Referring 
to the two tanks, Fig. 13, as A and B, assume tank A to be full 
of air at a pressure sufficient to sustain the back pressure or head 
of the discharge water column and tank B to be full of water. 
The air compressor is running and taking the compressed air 
out of A and passing it over into B. At this stage (the beginning 
of a cycle) no work is demanded of the air compressor except 




w Water Supply 

Fig. 13. 



that necessary to overcome friction in the air and water pipes, 
but as the air is exhausted out of A the compressor must raise 
the pressure to that of the constant water head. This recom- 
pressed air goes into B and forces the water out. At a certain 
period in the cycle the air pressure in A will have dropped to a 
point when water will begin to flow in through the intake valves. 
After this point in the cycle we may assume that for every volume 
of air taken out of tank A an equal volume of water flows in, thus 
maintaining a constant air pressure in A until the tank is filled 



SPECIAL APPLICATIONS OF COMPRESSED AIR 71 

with water. At this point the water will start up the air pipe and 
a sudden drop of pressure will occur in the intake pipe to the com- 
pressor. It is this sudden drop that is utilized to operate the 
reversing switch, which completes the cycle. 

From the foregoing it becomes evident that the mathematical 
analysis will involve the matter presented in Arts. 14 and 15, 
and there are two problems to solve for any installation: first, 
to determine the piston displacement of the compressor required 
to deliver a specified quantity of water per minute, say, second, 
to design the steam end of the compressor so as to meet the 
maximum demand for power which occurs once in each cycle. 

The first problem can be solved by Eq. (17), Art. 15, which 
may be modified thus: 
Let v a = the actual intake capacity of the compressor (usually 
about 70 per cent, of the piston displacement) ; this 
may be taken in cubic feet per minute. 
Let m = number of minutes required to bring the pressure 
down from p to p m . 

Then by Eq. (17) : 

log p - log p m 



m = 



log (7 + Va) - log 7 



7 being the volume of one tank and the air pipe between tank and 
switch, p and p m being the highest and lowest pressures re- 
spectively occurring in a tank in one cycle. If a tank full of water 
(volume 7) is to be delivered in n min. the time n measures the 
length of a cycle, and is divided into two parts: first, that just 
noted as m; and second, that required to draw the tank full of 
water after water begins to flow in under pressure pw. This 

7 
latter is — Hence 

Va 

, 7 

n = m H 

Va 

The solution must be made by trial. Thus assume v a and find 
m, then n by the equation next above. Repeat until a satis- 
factory n has been found. 

The second problem must be solved by the matter developed in 
Art. 14. There it is shown that, with sufficient accuracy for 
designing, the maximum rate of work occurs when r = 3. Hence, 
having determined v a , the maximum rate of work demanded of 



72 COMPRESSED AIR 

the steam end can be gotten by Eq. (8) or Table I with r — 3, 
due allowance being made for efficiency, etc. 

Since the air pipes have an effect analogous to the clearance 
space in engines they should be made small, even at some sacrifice 
in friction. A velocity of 40 to 50 per second may be allowed 
in the air pipes. 

The tanks should have a volume not less than ten times the 
volume in the air pipes. 

Theoretically, the pump tanks may be placed above the water 
supply as shown in Fig. 13, but the cycle can be shortened by 
submerging the tanks and thus increasing the pressure p m . In 
any case where the tanks are to be submerged the valves can 
still be placed above water and the water siphoned over into the 
tanks as suggested in Fig. 13a. 

The most important application of this return-air pumping 
system has been in pumping slimes, sand and acids — such material 
as would soon ruin a reciprocating or centrifugal pump. A 
number of such pumps are in use pumping cement "slurry" 
which is a fine-ground alkaline mud or slime occurring in the 
process of manufacture of Portland cement. 

The agency or force usually utilized for automatically operating 
the switch is the suction or partial vacuum in the intake pipe 
which (as the tanks are usually installed) will suddenly increase 
when water starts up the air pipe as described above. Other 
means that could be used to move the switch are : The difference 
in pressures in the intake and discharge pipes. This difference 
gradually increases until the switch is thrown and would be util- 
ized when the tanks are deeply submerged. Another means 
would be to switch after an assigned number of revolutions of 
the compressor — the number being that necessary to run a cycle 
as described above. 

The switch is usually made in the form of a piston valve. 
Details would be inappropriate in this volume. 

Example 36. — Design a return-air pump to deliver 200 cu. ft. 
per minute under a head of 100 ft.; the tanks to be placed at 
water level (as in Fig. 13a) and air pipes to be 500 ft. long. 

Solution. — The ratio po -J- p n is the same as the ratio of the 
water heads (taking amtospheric pressure as a head of 33.3 ft.). 
Then this ratio is 133.3 -f- 33.5 = 4. Assume for first trial that 
the tanks have a volume of 400 cu. ft. each, and that effective 



SPECIAL APPLICATIONS OF COMPRESSED AIR 73 



RETURN AIR PUMP 




Fig. 13a. 



74 COMPRESSED AIR 

intake capacity of the compressor is 1.5 cu. ft. per stroke. Then 
the number of strokes required in a cycle is 

400 log 4 

n = T? + 1 „m : i Ana = 266 + 370 = 636 ' 

1.5 log 401.5 - log 400 

636 strokes = 318 revolutions to deliver 400 cu. ft., or 159 
revolutions to deliver 200 cu. ft. per minute. 

This speed is excessive, but before we make another trial we 
will see what size air pipe will be necessary in order to prescribe 
the correct size of tanks. 

159 X 2 X 1.5 = 477 cu. ft. per minute intake to the com- 
pressor. This is the maximum rate of passage of air through 
either air pipe and occurs once in a cycle, and that just at the 
end when the pressure in the air pipe is about that of the atmos- 
phere. It is at this time that friction in the air pipe is greatest 
and we may allow a drop (/) of 5 lb. in order to economize in size 
of both air pipes and tanks. Then by formula (27d) 

/o.60 X 500 X (^PfY 

or by Plate III we see that a 3^-in. pipe will give a drop of 4 lb. 
in 500 ft. The volume in a 3^-in. pipe 500 ft. long is 33.5 cu. ft. 
Since we may use tanks having ten times the volume of the air 
pipe, we will recalculate for tanks 335 cu. ft. and a compressor 
intake capacity of 2 cu. ft. per stroke. Then 

n = ^T + i — oo 7 l0g 1 oo. = 167 + 235 = 402, 
2 log 337 — log 335 

whence 402 strokes or 200 revolutions deliver 335 cu. ft. and 
120 would deliver 200 — which is about the right speed for such 
a compressor. 

It remains to find the maximum rate of work required of the 
steam end of the compressor. The greatest ratio of compression 
occurring in a cycle is 4, but by Art. 14 we know that the great- 
est rate of work occurs when the ratio is about 3, and that this 
max. rate is 

w 144 poVa 144 X p V a _„ Tr 
W = —-3- = —j^sy- = 59poK, 

(ft) n - 1 V 



SPECIAL APPLICATIONS OF COMPRESSED AIR 75 

where po is the constant delivery pressure in pounds per square 
inch, and V a is the effective intake capacity of the compressor 
in cubic feet per minute or seconds as desired. If we take V a 
in cubic feet per second and divide by 550 we get horsepower. 
Then approximately the max. horsepower rate is one-tenth 
poV a - This is a general rule when r goes up to 3. 

Note that po is in pounds per square inch and that n is not the 
same as in the next preceding equation. 

Applying this to the numerals above we get 

,, , 133.3 X 0.434 X 120 X 2 X 2 ., 

Max. horsepower = - — nn w ... — - = 44. 

bO X 10 

A description of the installation of a return-air pump and a 
full discussion of the theoretic design can be found in Trans. Am. 
Soc. C. E., vol. 54, page 1, Date, 1905. 

Art. 37. Simple Displacement Pump. — First Known as the 
Shone Ejector Pump. — In this style of pump the tank is sub- 
merged so that when the air escapes it will fill by gravity. The 
operation is simply to force in air and drive the water out. When 
the tank is emptied of water, a float mechanism closes the com- 
pressed-air inlet and opens to the atmosphere an outlet through 
which the air escapes, allowing the tank to refill. Various 
mechanisms are in use to control the air valve automatically. 
The chief troubles are the unreliable nature of float mechanisms 
and the liability to freezing caused by the expansion of the escap- 
ing air. Some of the late designs seem reliable. 

The limit of efficiency of this pump is given by formula (18) 
and Table VI. The pump is well adapted to many cases where 
pumping is necessary under low lifts. In case of drainage of 
shallow mines and quarries, lifting sewerage, and the like, one 
compressor can operate a number of pumps placed where con- 
venient; and each pump will automatically stop when the tank 
is uncovered and start again when the tank is again submerged. 
See page 120 for design of a system of such pumps. 



CHAPTER VI 
THE AIR-LIFT PUMP 

Art. 38.— The air-lift pump was introduced in a practical way 
about 1891, though it had been known previously, as revealed 
by records of the Patent Office. The first effort at mathematical 
analysis appeared in the Journal of the Franklin Institute in 
July, 1895, with some notes on patent claims. In 1891 the 
United States Patent Office twice rejected an application for a 
patent to cover the pump on the ground that it was contrary to 
the law of physics and therefore would not work. Altogether the 
discovery of the air-lift pump served to show that at that late 
date all the tricks of air and water had not been found out. 

The air lift is an important addition to the resources of the 
hydraulic engineer. By it a greater quantity of water can be 
gotten out of a small deep well than by any other known means, 
and it is free from the vexatious and expensive depreciation and 
breaks incident to other deep well pumps. While the efficiency 
of the air lift is low it is, when properly proportioned, probably 
better than would be gotten by any other pump doing the same 
service. 

The industrial importance of this pump; the difficulty sur- 
rounding its theoretic analysis; the diversity in practice and 
results; the scarcity of literature on the subject; and the fact 
that no patent covers the air lift in its best form, seem to justify 
the author in giving it relatively more discussion than is given 
on some better-understood applications of compressed air. 

Art. 39. Theory of the Air-lift Pump. — An attempt at 
rational analysis of this pump reveals so many variables, and some 
of them uncontrollable, that there seems little hope that a satis- 
factory rational formula will ever be worked out. However, a 
study of the theory will reveal tendencies and better enable the 
experimenter to interpret results. 

In Fig. 14, P is the water discharge or eduction pipe with 
area a, open at both ends and dipped into the water. A is the 

76 



THE AIR-LIFT PUMP 



77 



air pipe through which air is forced into the pipe, P, under pres- 
sure necessary to overcome the head D. 6 is a bubble liberated 
in the water and having a volume which increases as the bubble 
approaches the top of the pipe. 

The motive force operating the pump is the buoyancy of the 
bubble of air, but its buoyancy causes it to slip through the 
water with a relative velocity u. 

In one second of time a volume of water = au will have passed 
from above the bubble to below it and in so doing must have 
taken some absolute velocity s in passing the contracted section 
around the bubble. 

Equating the work done by the buoyancy of the 
bubble in ascending, to the kinetic energy given 
the water descending, we have 



wOu = wau ^— where w — weight of water, 
2g 



or 







2g 



(a) 



■— -Qji' 



Fig. 14. 



~— is the equivalent of the head h at top of the 
pipe which is necessary to produce s, therefore 

a 
Suppose the volume of air, 0, to be divided 
into an infinite number of small particles of air, 
then the volume of a particle divided by a would be zero and 
therefore s would be zero; but the sum of the volumes (=0) 
would reduce the specific gravity of the water, and to have a 
balance of pressure between the columns inside and outside the 
pipe the equation 

wO = wah 
must hold. 

Hence again h = -, showing that the head h depends only 

CI 

on the volume of air in the pipe and not on the manner of its 
subdivision. 

The slip, u, of the air relative to the water constitutes the chief 
loss of energy in the air lift. To find this apply the law of physics, 
that forces are proportional to the velocities they can produce in a 
given mass in a given time. The force of buoyancy wO' of the 



78 COMPRESSED AIR 

bubble causes in 1 sec. a downward velocity s in a weight of water 
wan. Therefore 

wO s 

wau g 
Whence 

a s 2 

u == - — But — = 7>~ as proved above. 
as a 2g 



Therefore 



2 \o 2 



This shows that the slip varies with the square root of the 
volume of the bubble. It is therefore desirable to reduce the 
size of the bubbles by any means found possible. 

If u = ~> then the bubble will occupy half the cross-section of 

the pipe. This conclusion is modified by the effect of surface 
tension, which tends to contract the bubble into a sphere. 
The law and effect of this surface tension cannot be formulated 
nor can the volume of the bubbles be entirely controlled. Unfor- 
tunately, since the larger bubbles slip through the water faster 
than the small ones, they tend to coalesce; and while the conclu- 
sions reached above may approximately exist about the lower end 
of an air lift, in the upper portion, where the air has about re- 
gained its free volume, no such decorous proceeding exists, but 
instead there is a succession of more or less violent rushes of air 
and foamy water. 

The losses of energy in the air lift are due chiefly to three 
causes: first, the slip of the bubbles, through the water; second, 
the friction of the water on the sides of the pipe; and third, 'the 
churning of the water as one bubble breaks into another. As 
one of the first two decreases the other increases, for by reducing 
the velocity of the water the bubble remains longer in the pipe 
and has more time to slip. 

The best proportion for an air lift is that which makes the sum 
of these two losses a minimum. Only experiment can deter- 
mine what this best proportion is. It will be affected somewhat 
by the average volume of the bubbles. As before said, any 
means of reducing this volume will improve the results. 

Art. 40. Design of Air-lift Pumps. — The variables involved 
in proportioning an air-lift pump are: 



THE AIR-LIFT PUMP 



79 



1 

ff 



ti 






ff 



L 
r 



f 
1 

1 



Q = volume of water to be lifted, per second, 

h = effective lift from free water surface outside the discharge 

pipe, 
I = D + h = total length of water pipe above air inlet, 
D = depth of submergence = depth at which air is liberated in 

water pipe measured from free water surface outside the 

discharge pipe, 
v a = volume per second of free air forced into well, 
a = area of water pipe, 
A = area of air pipe, 
= volume of the individual bubbles. 

The designer can control A, a, D + h and v a but he has little 
control over 0, and cannot foretell what D and Q 
will be until the pump is in and tested. 

When the pump is put into operation the free 
water surface in the well will always drop. What 
this drop will be depends first on the geology and 
second on the amount, Q, of water taken out. In 
very favorable conditions, as in cavernous lime 
stone, very porous sandstone or gravel, the drop 
may be only a few feet, but in other cases it may 
be so much as to prove the well worthless. In any 
case it can be determined by noting the drop in 
the air pressure when the water begins flowing. 
If this drop is p lb., the drop of water surface in 
the well is 2.3 X p ft. 

Unless other and similar wells in the locality 
have been tested, the designer should not expect 
to get the best proportion with the first set of pip- 
ing, and an inefficient set of piping should not be 
left in the well. 

The following suggestions for proportioning air 
lifts have proved safe in practice, but, of course, 
are subject to revision as further experimental 
data are obtained (see Figs. 16 and 17). 

Air Pipe. — Since in the usually very limited space high veloci- 
ties must be permitted we may allow a velocity of about 30 ft. per 
second or more in the air pipe. 

Submergence. — The ratio n _,_ ^ is defined as the submergence 

ratio. 



1 



Fig. 15. 



D + h 
Experience seems to indicate that this should be not less 



80 COMPRESSED AIR 

■> 

than one-half; and about 60 per cent, is a common rule of 
practice. Probably the efficiency will increase with the ratio of 
submergence, especially for shallow wells. The cost of the extra 
depth of well necessary to get this submergence is the most 
serious handicap to the air-lift pump. 

v 
Ratio jY-' — -When air is delivered into water under submergence 

D its ratio of compression will be 

D + 33.3 



r = 



33.3 



33.3 ft. being a fair average for water head equivalent to one 
atmosphere. 

When air is mixed with water as in these pumps, it may be 
assumed to act under isothermal conditions. Then the energy 
in the air is p a v a log e r while the effective work realized by water 
delivered at top of discharge pipe is 



wQ i h+ B 



s being the velocity of discharge. Write ^— = Hi. Then if E 

be the efficiency of the pump (reckoned from energy in air deliv- 
ered) we have the equation 

wQ (h+hi) = E p a v a log e r. 
Whence 

Va w(h + hi) ..' 

Q ~Ep a log e r ^ 

In case of a pure-water air-lift pump w = 62.5, and we may 
take for average atmospheric conditions p a = 2,100. Then 
multiplying by 2.3 and using common logarithms the formula 
becomes 

th , v a h -\- Hi /0 _v 

For pure water ^ = 773ElogiQr (35) 

Complete data on several apparently well-designed air-lift 
pumps with ratio of submergence between 50 and 65 per cent, 
and total submergence between 350 and 500 ft. show E to have a 
value between 45 and 50 per cent, (see Art. 43). 

If we take E = 45 per cent., Eq. (35) becomes 

v a h + hi 



Q 35 logio r 



(36) 



THE AIR-LIFT PUMP 



81 









t 


— 
1 


-H- 




■ 


— — 




~T- 


: 


■: 


J 


:■ :: : 




■ jTT 






— 1 — 1 — 1 — 1 

1 




;; 


:. : :.:: 






— " 






- vo --rtiTif" 

. co . .... j . 

: cr 1 :TTr> -■■; z 

- W - ||j7. I 1 ~ 


itti;:: 




\ 




\ 




—— 


;.;■ 


- ; 


::: 


..:. 


— 




-* D r/5 -— — — — 


c, 

^ ' , i o 

1-1 

--- o 

-H — i-i->-- o 


".... 


"\ 


\ 




-T^V 


\ 


- -\ 




— rr 






:::: 


u " 1 1 L 1 1 ; 1 . 

u 


o 

— 1 h- o 

oo 


- 


\ 






\ 


:::.:: 


\ ■;, 








.:: :.. 


rtp — p-l= 


. : :; 


3* 






V 






v ^?\ 


73" 


fa\! 








o 

— ■ — + o 

to 














^b 


:\iV 




: tA 








' o 
. — . L. o 












<$■ 




V\ 












— — F" © 














\ 




\ 




v : 




^__ _.'..:.. : ' .;•: . 


,::; ' 


SE 


~ ■ 








~~~r~ 
----]■ 


._„_:_: 


IniL 




j-- 


\ 
\ 

\ 


V 




. : o 

— -— — - © 

: : : : n 

■ 
o 




: 


--p-- : 

; ; ■ 


- 




... 


■ 


:; 






J_ 






fun 


M^ 


rtTT + T: o 
1— L o 

-irMlilllo 



t- cs q,o 



Plate IV. 



82 COMPRESSED AIR 

Formula (36) is recommended -for the design of deep-well 
pumps. In this hi may be taken as about 6 ft. which is assum- 
ing a discharge velocity between 20 and 25 ft. For shallow wells 
hi may be taken as 1, which would correspond to a velocity of 
8. ft. 

The curves on Plate IV represent Eq. (36) for ratios of sub- 
mergence as shown thereon. Note that in Plate IV an efficiency 
of 45 per cent, is assumed. When more data have been collected, 
some modification may be found desirable. 

In any case some excess air capacity should be provided; for 
should the free water surface in the well drop more than antici- 
pated, after prolonged pumping, more air will be needed to 
maintain the discharge. This is apparent on Plate IV. Note 

v 
that as submergence ratio S decreases ^ increases. 

Velocity in the Water Pipe. — This is the factor that most affects 
the efficiency, but unfortunately, owing to the usual small area in 
the well, the velocity cannot always be kept within the limits 
desired. The complicated action and varying conditions in a 
well make the designer entirely dependent on the results of ex- 
perience in fixing the allowable velocities in the discharge pipes. 

The velocity of the ascending column of mixed air and water 
should certainly be not less than twice the velocity at which the 
bubble would ascend in still water. This would probably put the 
most advantageous least velocity in any air lift at between 5 and 
10 ft., and this would occur where the air enters the discharge 
pipe. 

The velocity at any section of the pipe will be 

Q + v 

s =-Z » 

a 

where Q and v are the volumes of water and air respectively and 
a the effective area of the water pipe, s increases from bottom to 
top probably very nearly according to the formula 

a \ ri I 
where 

K = increment of velocity, 
r — ratio of compression under running conditions, 
I = total length of discharge pipe above air inlet, 
x = distance up from bottom end of air pipe to section where 
velocity is wanted. 



THE AIR-LIFT PUMP 83 

The formula is based on the assumption that the volume of 
air varies as the ordinate to a straight line while ascending the 
pipe through length I. As the volume of each bubble increases 
in ascending the pipe, the velocity of the mixture of water and air 
should also increase in order to keep the sum of losses due to slip 
of bubble and friction of water a minimum; but for deep wells 
with the resultant great expansion of air the velocity in the upper 
part of the pipe will be greater than desired, especially if the dis- 
charge pipe be of uniform diameter. Hence, it will be advanta- 
geous to increase the diameter of the discharge pipe as it ascends. 
The highest velocity (at top) probably should never exceed 20 
ft. per second if good efficiency is the controlling object. 

Good results have been gotten in deep wells with velocities 
about 6 ft. at air inlet and about 20 ft. at top (see Art. 43). 

Figure 16 shows - the proportions and conditions in an air lift 
at Missouri School of Mines. 

The flaring or slotted inlet on the bottom should never be 
omitted. Well-informed students of hydraulics will see the 
reason for this, and the arguments will not be given here. 

The numerous small perforations in the lower joint of the air 
pipe liberate the bubbles in small subdivisions and some advan- 
tage is certainly gotten thereby. 

No simpler or cheaper layout can be designed, and it has 
proved as effective as any. It is the author's opinion that 
nothing better has been found where submergence greater than 
50 per cent, can be had. 

Art. 41. The Air Lift as a Dredge Pump. — The possibilities 
in the application of the air lift as a dredge pump do not seem 
to have been fully appreciated. This may be due to its being 
free from patents and therefore no one being financially interested 
in advocating its use. 

With compressed air available a very effective dredge can be 
rigged up at relatively very little cost and one that can be adapted 
to a greater variety of conditions than those in common use, 
as the following will show. 

Suggestions. — Clamp the descending air pipe to the outside 
of the discharge pipe. Suspend the discharge pipe from a derrick 
and connect to the air supply with a flexible pipe (or hose). 
With such a rigging the lower end of the discharge pipe can be 
kept in contact with the material to be dredged by lowering from 
the derrick; the point of operation can be quickly changed within 



84 COMPRESSED AIR 

the reach of the derrick, and the dredge can operate in very- 
limited space. In dredging operations the lift of the material 
above the water surface is usually small, hence a good submer- 
gence would be available. The depth from which dredging could 
be done is limited only by the weight of pipe that can be handled. 

In case of air-lift dredge pumps the ratio of submergence may 
be large and the weight per cubic foot lifted will be greater than 
62.5 lb. In case heavy coarse material (such as gravel) is being 
lifted, the velocity should be high. 

Though the author has found no data by which the efficiency 
of such pumps can be determined, the following example is 
taken for illustration. 

v 
Example 35. — What is the ratio -^- for a dredge pump when 

submergence = 30 ft., net lift = 6 ft., velocity at discharge 
= 12 ft., percentage of silt = 33.3 per cent., weight of silt = 
100 lb. per cubic foot, and efficiency = 0.333? 

Solution. — Referring to Eq. (34), w becomes 75, hi = 2, 
r = 1.9 (submergence 30 ft. in pure water). Whence we get 

Va_ = 75 X 8 + (100 - 62.4)^ X 30 

Q ' 0.333 X 2,100 X 2.3 X 0.279 ' 

Note that the second term in the numerator is the work done in 
lifting the silt through the 30 ft. of water. 

Art. 42. Testing Wells with the Air Lift.— The air lift affords 
the most satisfactory means yet found for testing wells, even if 
it is not to be permanently installed. Such a test will reveal, in 
addition to the yield of water, the position of the free water 
surface in the well at every stage of the pumping, this being shown 
by the gage pressures. However, some precautions are neces- 
sary in order properly to correct the gage readings for friction 
loss in the air pipe. 

The length of air pipe in the well and any necessary correc- 
tions to gage readings must be known. 

The following order of proceeding is recommended. 

At the start run the compressor very slowly and note the pres- 
sure pi at which the gage comes to a stand. This will indicate 
the submergence before pumping commences, since there will 
be practically no air friction and no water flowing at the point 
where air is discharged. Now suddenly speed up the compressor 
to its prescribed rate and again note the gage pressure p 2 before 



THE AIR-LIFT PUMP 



85 



any discharge of water occurs. Then p 2 — Vi = P/ is the pres- 
sure lost in friction in the air pipe. When the well is in full 
flow the gage pressure p s indicates the submergence plus friction, 
or submergence pressure p s = p 3 — p/. The water head in feet 
may be taken as 2.3 X p s . Then, knowing the length of air 
pipe, the distance down to water can be computed for conditions 
when not pumping and also while pumping. 



./£=> 




Fig. 16. 




Art. 43. Data on Operating Air Lifts. — In Figs. 16 and 17 are 
shown the controlling numerical data of two air lifts at Rolla, 
Mo. These pumps are perhaps unusual in the combination of 
high lift and good efficiency. The data may assist in designing 
other pumps under somewhat similar circumstances. 

The figures down the left side show the depth from surface. 



86 COMPRESSED AIR 

The lower standing-water surface is maintained while the pump 
is in operation; the upper where it is not working. 

The broken line on the right shows, by its ordinate, the vary- 
ing velocities of mixed air and water as it ascends the pipe. 

The pump, Fig. 16, delivers 120 gal. per minute with a ratio 

j'l'pp O 1 T» 

: — = 6.0. The submergence is 58 per cent, and efficiency 

net energy in water lift 

= ; — = 50 per cent. 

pv log e r 

The pump, Fig. 17, delivers 290 gal. per minute with a ratio 
, = 5.2. Submergence = 64 per cent, and efficiency = 

net energv in water lift 

—, = 45 per cent. 

pv log e r 

The volumes of air used in the above data are the actual 
volumes delivered by the compressor. The volumetric effi- 
ciencies of the compressors by careful tests proved to be about 72 
per cent. 



CHAPTER VII 

RECEIVERS AND STORAGE OF ENERGY BY 
COMPRESSED AIR 

Art. 44. Receivers for Suppressing Pulsations Only. — Every 
air compressor bf the reciprocating type has, or should have, an 
air receiver on the discharge pipe as close as possible to the com- 
pressor outlet. The chief duty of this receiver is to absorb or 
take out the pulsations caused by the intermittent discharge 
from the compressor in order that the flow of air through the 
discharge pipe (beyond the receiver) may be uniform, a condition 
evidently essential to efficient transmission. Incidently the 
receiver serves as a separator for some of the oil and water in the 
air and as a store of compressed air that may be drawn from when 
the demand is temporarily in excess of the compressor delivery. 

There is no standard rule, nor can there be one, for proportion- 
ing these receivers. However, the service demanded of the air 
will usually indicate whether or not a large receiver is desirable. 
The least size would apply in cases where the use of the air is 
continuous and the needed pressure constant — as in air-lift 
pumps. In such cases the requirement of the receiver is solely 
to suppress the pulsations. For such cases a rational formula 

v 
for the volume of the receiver would be V = c - in which V is 

r 

the volume of the receiver, v the piston displacement of one stroke 
(of low-pressure cylinder), r the total ratio of compression, and 
C an empyrical coefficient fixed by experiment or observation. 

Observe that v -f- r is the volume of compressed air per stroke, 
which being suddenly discharged causes the pulsations. The 

V ' 

question remains, what ratio of V to - will be necessary to suffi- 
ciently suppress the pulsations? The author finds no light on the 
subject in compressed air literature. Practice does not seem to 
distinguish between this case and the more usual one where 
some storage capacity is needed. The author is of the opinion 
that in such cases (the air lift for instance) a much smaller re- 

87 



88 COMPRESSED AIR 

ceiver could be used without detrimental effect, and thereby the 
cost reduced. 

Art. 45. Receivers. Some Storage Capacity Necessary. — 
In the majority of compressed-air installations the use of, or 
demand for, air will not be constant, as for instance in machine 
shops, quarries, mines, etc. In any such case the use of air may 
exceed the compressor capacity for a short time and then for a 
time may not be as great as the compressor capacity. In these 
(the more common) cases the receiver is intended to serve as a 
storage reservoir in addition to its several other duties. 

The problem of determining the necessary volume for the stor- 
age is simple, provided the maximum rate of use and its duration 
in time can be gotten ; but this is seldom possible as will be readily 
conceded when the complexity and irregularity of the service 
is considered. 

However, the problem may be better understood if expressed 
as a formula: 
Let V = volume of receiver, or storage reservoir, in cubic feet, 
v a — free-air capacity of compressor in cubic feet per min., 
Pi = highest pressure (absolute) permitted in the system, 
7> 2 = lowest satisfactory working pressure permissible, 
R = maximum rate of usage, cubic feet free air per min., 
T = duration in minutes through which R is continued. 

To get a simple and sufficiently approximate relation assume 
isothermal changes and equate the pressure by volume products 
thus: 

PlV + TpaVa = P2V + TRp a . 

Whence, 

V = Pa T{R - v a ) 
Pi- Pi 

Example. — A shop has a compressor with v a = 300 and normal 
pressure p\ = 100 ft. A drop of 10 lb. (p 2 = 90) is permissible 
when the demand is double the average for 2 minutes. Then 

1 f; 

V = ~ X 2 X (600 - 300) = 900. 

A calculation, such as above, applied to the more common in- 
stallations, will show the desirable receiver capacity much greater 
than is installed. 

The common practice seems to be to install a compressor 



RECEIVERS AND STORAGE OF ENERGY 89 

capable of meeting the maximum demand without storage, and 
then let it run idle much of the time. Going to this extreme is 
profitable for the compressor makers, but expensive to the user 
in first cost of the compressor and still more so in the continual 
cost of extra fuel to run the larger compressor even though idle. 

Where a compressor has been installed with inconsiderable 
receiver (or storage) capacity and the business outgrows the 
capacity of the compressor as thus equipped, the addition of a 
considerable storage volume may defer the time for purchase of 
a larger compressor for several years, and at the same time get 
the needed additional air more economically than if a larger 
compressor were installed. This claim of economy is based on 
the fact that a small machine running more continually and with 
a nearly constant load is more economical than a larger machine 
running constantly but with intermittent load. The case is 
analogous to the use and duty of a distributing reservoir in a 
water-supply system. 

Art. 46. Hydrostatic Compressed-air Reservoirs. — In cases 
where it is desired to store large units of energy in the form of 
compressed air, and that energy is to be drawn out with but little 
drop in the air pressure, a computation of the volume necessary 
for tanks under conditions heretofore assumed is discouraging. 

Example. — Assume that storage is to be provided for air at 

125 lb. (absolute) such that 100 hp. can be drawn from the storage 

for 1 hr. with a drop in pressure to 100 lb. What volume of 

storage is needed? For this example assume all changes to be 

isothermal. 

lot: inn 

Then 100 X 33,000 X 60 = p a V a log e ^| - p a V a log e ^ 

= 2,116 V a (2.140 - 1.917), 

whence V a = 4,200,000 and V = 50,000 approximately. Sup- 
pose now that all the compressed air in volume V at pressure 
125 can be used without any reduction of pressure. What 
volume would give 100 hp. for 1 hr.? 

Then 100 X 33,000 X 60 = p a V a log 8.5, 

whence V a = 43,700 and V = 5,150 or about one-tenth of that 
required under the first assumption. 

This latter condition (making the entire volume of compressed 
air available without reduction of pressure) can be accomplished 



90 



COMPRESSED AIR 



simply and economically by the scheme illustrated by Fig. 18 
which needs but little explanation. 

The water head against the air may be assumed constant. 
The dip down in the water pipe below the air reservoir is to 
prevent blowout through the water pipe should all the water be 
forced out of the air reservoir. A popoff, or an automatic, stop 
for the compressor, would be adjusted to act when the water line 
dropped down below the air tank as at C. Evidently the water 
pipe ABC need not be vertical nor in a vertical plane. The water 
reservoir can be economically placed on a hilltop in the neighbor- 




Fig. 18. 



hood, or the air reservoir can be placed in underground chambers 
(abandoned chambers in mines, for instance) and the water 
reservoir at the surface. 

This last suggestion naturally leads to that of using under- 
ground chambers naked, that is, without the steel tanks. This 
is quite feasible. To make the walls of such a chamber tight 
against escape of air into the rock the " cement-gun" is ideal. 
Note that there is no necessity for smooth or even surfaces. 
The cement-gun may be found more efficient if used while the 
chamber is under some pressure. The cement will thus be driven 
into every crevice and pore into which air may tend to escape. 



CHAPTER VIII 
FANS 

Art. 47. — The discussion in this article will apply to any 
centrifugal pump and to fans of the low-pressure type such as 
are applied in ventilation of mines and buildings, when change 
in density of the air may be neglected. 

Though the discussion is brief, the student entering the subject 
for the first time will find some difficulty in keeping in mind the 
several qualifying conditions such as relative velocities, velocity 
heads, pressures within and without the wheel, etc. He is warned 
not to jump at conclusions in this nor any other branch of fluid 
mechanics, but is urged to study and review each demonstration 
over and over again until familiar with it. We hope to encourage 
an interest in this subject by saying beforehand that there are 
several fallacious opinions that can be successfully contended 
with only by those who are well grounded in the following under- 
lying theory. 

We will first study the theory as revealed by the laws of pure 
mechanics and the conservation of energy, without considering 
the effects of friction, imperfection of design or improper opera- 
tion. Formulas thus obtained will not closely check with re- 
sults from a pump or fan in operation, but they tell what per- 
fection would be, and so show how far short we fall in practice; 
and they point to the best lines of improvement in design and in 
operation. 

In addition to the symbols shown on Fig. 19, the following will 
be used: 

w = weight of cubic unit of fluid, 

b = width of discharge at outer limit of vanes, 

61= width of inlet at inner limit of vanes, 

Q = volume of fluid passing, cubic feet per second unless 

otherwise stated, 
W = total weight of fluid passing per second wQ, 
p = pressure head immediately after escape from revolving 

wheel, 
H = total head given to the fluid by the wheel. 

91 



92 



COMPRESSED AIR 



The reason for using heads instead of pressures is that the 
formulas are thereby simplified. Note that head must be in 




C cdx 



B 
1 



mm^m 




Fig. 19. 



feet of the fluid passing. In case of air the air head is to be of 
constant density. 



FANS 93 

Art. 48. Purely Centrifugal Effects. — Consider a prism of fluid 
of unit area and extending between the limits r\ and r, in a 
revolving wheel without outlet, as CB, Fig. 19. The centrifugal 
force of an elementary disk across this prism, of thickness, dx, 
and distance x, from the axis of revolution, is by well-known laws 
of mechanics 

7/ . wu x 2 dx 

df = 

gx 

where u x is the velocity of revolution at the distance x from the 
center. Thus 



3/ 

u x = - u 
r 



and therefore 



wu 



df = — r xdx 

gr 2 



and the total centrifugal force/, which is effective at the outer end 
of this prism, is the integral between the limits x = r and x = rj. 
Hence 

wu 2 ,-„ „ N w 



2 gr 2 2 g 



since 



Mi = — U. 

r 

This is the pressure on a unit area at the circumference of the 
wheel, and, evidently, it is independent of the form or cross- 
section of the arm CB. Now, pressure divided by weight gives 
head. Hence, the pressure head against the walls of the wheel 
at the circumference is 

7 . u 2 — Mi 2 
•v h 1 = 



2g 

Note that if ri = then h = „ — 

2g 

Note that this h does not include velocity of rotation. 

Now, if an orifice be opened at the circumference, in any di- 
rection whatever, and the pressure outside be the same as at the 
entrance, the velocity of the discharge, relative to the revolving 
walls of the wheel, will be 

V = V2~gh or V 2 = u 2 - u x \ 

Note that if r = 0, V =-u. 

Note also that the absolute velocity v of discharge is made up 



94 COMPRESSED AIR 

of the two components V and u and in amount v 2 = u 2 + V 2 
+ 2 uV cos /3 = 2 u 2 + 2 wF cos when ri = 0. 

v 2 
Total head, i/, in the departing fluid = ~— or 

H _ u 2 + uVgos fl , _. 

When there is a discharge, there must be an initial velocity, 

Vi, at the entrance, and this must be considered in the final 

head within the wheel. Thus, the total relative head at B will 

now be 

7 Vi 2 , u 2 — Ui 2 
h = ^ h 



2flf ' 2g 

and the velocity of the discharge, relative to the revolving parts, 
will have the relation 

v 2 = y, 2 + u 2 - wi 2 (i) 

Suppose, now, that CB is a radial frictionless tube, open at 
both ends, and that a particle of matter starts from a state, Vi, 
relative to the tube, and moves out, without change of pressure, 
from radius r x to r, in obedience to the laws of centrifugal force 
(or acceleration). Its radial acceleration, when distant x from 
the center, is, by well-known laws of mechanics : 

Acceleration 

u x 2 _ dV x 
x dt 

Also 

Vx = dt' 
Therefore, by eliminating dt, we get 

u x 2 dx 



but, as before, 



V x dV x = 

x 



X 
U x = — U 

r 



(sub x indicating the conditions at the distance x from the cen- 
ter) . Therefore, 

V x dV x = —r xdx. 



FANS 



95 



Integrating between the limits V and Vi, on one side, and r and 
ri, on the other, we get as before 



7 2 -Fi 2 = U 2 _ Mi s 



(I) 



Art. 49. Impulsive or Dynamic Effects. — We have now to 
study the effect of picking up the fluid at entrance to the moving 
parts of the wheel. This will be studied by a method somewhat 
different from that preceding: 

Assuming the fluid to be at rest until influenced by the wheel, 
we see that during each second there is a weight, W, brought to a 




velocity, v, Fig. 20. Now the reaction against the wheel due to 

Wv 
the creation of the velocity, v, is F = — and the component of 

this velocity opposite in direction to the rotation, u, is v cos 
and this equals u — V cos (180° — 8) = u + V cos /3 and since 
work is force multiplied by distance, the work done in overcoming 
the reaction is 

W W 

u(u -f- V cos 13) — = (u 2 + uV cos 8) — 

g g 

If H be the total head given the fluid up to the point considered, 



96 COMPRESSED AIR 

then work done = WH, since all the head has been imparted by 
the wheel. 
Hence, 

H = W<1 + uV cos A (37) 

9 

Note that it is preferable to use the angle /3 rather than a for /3 
is fixed and is an element in the design of the machine, while a 
varies with u and V. 

The demonstration above evidently applies, however short the 
radial depth of the vanes (r — r{). So we may say that it 
applies at the entrance where r — r x = 0. Here, then, we find Eq. 
(37) applies in case of purely impulsive, or dynamic action with 
neither centrifugal force nor centrifugal acceleration. 

It is now apparent that no matter at what distance from the 
center of rotation the fluid is engaged by the wheel, it will have 
imparted to it a head the same as if it had been under influence of 
the wheel from the center out. 

Art. 50. Discharging against Back Pressure. — Note that so 
far we have assumed that the pressure against the discharge is 
the same as at intake. Under this condition the relative velocity 
of escape will be V = u, no matter in what direction V may be, 
relative to the wheel. 

We have now to establish a general formula for H when pres- 
sure head against outlet is p. and at inlet p\. Note that pi may 
be and generally is negative in centrifugal pumps, but in fans 
it is usually zero. 

We have established the fact that the pressure head developed 

within the wheel, when no discharge is allowed, is h = ^— ■ Now 

i & J 2 g 

if an orifice be opened through the periphery of the wheel into 
the discharge duct where pressure head = p, the velocity of 
escape (relative) will be 

V 2 = 2g(~+p i -p)=u* + 2g(p 1 -p) (II) 

Whatever the absolute velocity of escape v may be, the total 
absolute head added to the fluid by the machine will be 

H=£g + V-Vl (Hi) 

Note that when pi is negative (or suction), it becomes a plus 
addition in (III). 



FANS 97 

From pure trigonometry we have in any case 

v 2 = u 2 + V 2 + 2 uV cos j(3. 

Now in (III) replace v 2 from the last expression, then replace 
V 2 with its value from (II) and we get as before 

H = u 2 + uV cos ft , 

g 

We have now proven Eq. (37) to be correct for both purely 
centrifugal and impulsive action, and to be independent of 
entrance and discharge pressures. 

Equations (II), (III) and (37) are the theoretic relations when 
effects of friction and imperfections of design are neglected. 
Results in practice may be, and often are, quite different from 
what this theory would indicate, due to imperfections of design, 
some of which cannot be overcome entirely. 

v 2 
Note that if p\ = P, then H = ^— and V = u; and if pi = p 

in 2 
and = 90° then v 2 = 2u 2 . When = 90°, H = — irrespective of 

u 2 
pressures. Also, when (3 is less than 90°, H is greater than — 

u 2 
and when /3 is greater than 90°, H is less than — • 

The pressure that a pump or fan can produce depends on H. 
P = wH when the whole energy is transformed into pressure 
head. Otherwise, in general 

at or near outlet, friction being neglected. 

The work required of the machine (neglecting friction, etc.) 
is WH = wQH where W and Q are total weight and total volume 
passed, respectively, and this will in actual performance nearly 
equal the theoretic, regardless of friction and other losses. 

qi 2 

Note that V may be zero and still H = — . In this case the 

9 

u 2 
fluid revolving in the wheel has a pressure head = ^— , and a ve- 

u 2 
locity head = „— , the total head being the sum. In this case 

the work is zero since W = 0. If the pressure head, p, in the 

discharge duct be ^— , there will be no discharge, the pressure 

^g 

7 



98 COMPRESSED AIR 

inside and outside the wheel balancing. As p decreases V in- 
creases (see Eq. (II)) and therefore also Q increases. 

In case of pumps when /3 is less than 90° the pump cannot start 
a discharge under full back pressure, but if /3 be less than 90° 
the pump may start under its full head. 

Art. 51. Designing. — The dimensions of any pump or fan must 
conform to the following formulas, which hold in all cases: 

Q = 2 wrbV sin ,3 = 2 vrb sin p^u 2 - 2 g(p x - p) (V a ) 

Also Q = 2 TTibiVi sin (V b ) 

Note that V sin (5 and Vi sin 8 are the radial components of 
velocity of discharge at outlet and inlet respectively. 

In designing a fan or pump, the chief factors are H and Q. 
By equation (37) these factors are seen to be interdependent 
(except where j8 = 90°), since for any completed machine Q is 
directly proportional to V. 

Unfortunately there seems no rational formula for V. The 
formula V 2 = u 2 — 2 g(pi — p) is theoretically correct, but there 
is no satisfactory way to determine or fix p in this formula pre- 
liminary to the design. This fact necessitated dependence on 
the cut-and-try process by the pioneers in this field; though now 
data have been gotten for so many pumps and fans of various 
styles, showing the relation between head and discharge, that 
designers can proceed with tolerable confidence except where 
some radical departure in design is to be tried (see Art. 53) . 

Assuming that the designer has the data showing relation 
between H and Q (or V) for the style of machine he is going to 
copy, he has equations V a and Vb to conform to. 

The angle /3 should be selected with due regard to the service 
required of a machine and method of propulsion. Notice that, 
assuming u constant, when : 

Angle j8 is less than 90°, H increases as Q increases. 

Angle fi is equal to 90°, H is independent of Q. 

Angle j(3 is greater than 90°, H decreases as Q increases. 

It is common to assume the fluid as approaching the inner 
limits of the fan blades in a radial direction (see Fig. 19) even 
when no guide vanes are provided, though in that case the as- 
sumption may be far from the truth. 

Note also that with H fixed, u must increase as /3 increases. 
This fact is taken into consideration when a machine is to be 



FANS 



99 



driven by a high-speed electric motor or a steam turbine. In 
such cases the embarrassing condition in the design is to apply 
the high rotative speed without getting excessive head; hence, 
in such cases the angle /3 is made greater than 90°. 

Another advantage in turning the vanes backward (0 greater 
than 90°) in case of electric-driven machines, is that the machine 
will not be overloaded when the head or resistance is suddenly 
thrown off, with the resulting great increase in discharge (which 
increases V) (see Fig. 21). 

In cases where a machine is to be run by a reciprocating 
steam engine direct-connected and the designer has trouble in 



*9 



07 



24 

























































































































































































































f? 


<90° 
























SI 


LUt-C 


ffH 


sad 


















^^ 


■^.^ 


























ft' 


= 90° 














































































^ 


'^90 


> 








"^> 


N 








































's 


N 


































\ 


s 















123456789 10 

Cubic Feet per Minute -Thousands 

Fig. 21. — Characteristic curves. Constant speed. Varying discharge. 

getting the desired head with the limited speed, he will find it 
advantageous to turn the vanes forward. 

Where a constant head (or pressure) is desired with varying 
quantity as in sewage pumps and ventilating fans for buildings, 
the most rational design would be to provide an adjustable dis- 
charge with radial vanes. 

Another condition that should receive consideration in de- 
signing, or selecting, a machine is where a pump is to force water 
through a long pipe and where a fan is to force air through a mine. 
In such cases the greater portion of the resistance to be over- 
come is friction head, and it is well known that this varies with 
the square of the velocity, and, therefore, any increase in the 



100 COMPRESSED AIR 

quantity will be accompanied with a relatively greater resisting 
head. This case would be best met by setting the vanes radial 

(at discharge). Then, theoretically, H = — and quantity 

varies directly with u, when running under most favorable con- 
ditions. Now, as stated before, friction will vary as quantity 
squared. Hence, H will vary directly with the friction. This 
very nearly meets the most desired condition in mine ventilation 
where practically all the resistance is due to friction. To 
illustrate numerically: Suppose it is decided to double the 
quantity of air passing through a mine. If we double the speed 
of the fan we get double the flow of air, four times the pressure, 
four times the friction and eight times the power will be neces- 
sary to run the fan. 

Probably the engine or motor in the above example would be 
incapable of developing eight times its normal power. How 
then can the problem be solved? Will it be effective to install 
a duplicate of the first fan and discharge both into the mine? 
No, for we would be trying to put through double the quantity 
without increasing the pressure. The result would be a some- 
what greater pressure and quantity, but both fans would now 
be working inefficiently, if they were properly adopted to the 
first condition. 

Would the problem be solved by placing another fan in series 
(or tandem) with the first? No, for now we would be proposing 
twice the head with no increase of volume. There would be some 
increase in volume, but not double, and again the fans would 
not be working under best conditions. 

By this simple example it is evident that if there is a radically 
different quantity to be sent through a long conduit (pipe, flume 
or mine), the only scientific solution is to install a new machine 
adapted to work efficiently under the new conditions. 

Art. 52. Testing. — The following discussion refers to fans or 
blowers. 

Manufacturers of recognized standing have facilities for testing 
their machines and should know, with sufficient accuracy for 
commercial purposes, what their machines will do and the condi- 
tion under which they will work most efficiently, and the pur- 
chaser of a machine for any important serivce should demand a 
guaranteed performance chart for the machine, this chart to give 
information equivalent to that shown on Fig. 22. Then, in the 



FANS 101 

acceptance test the engineer for the purchaser might be content 
with checking a few points on the performance chart under con- 
ditions approximating those under which the machine is to work. 

In the purchaser's test, the data wanted are quantity, head and 
efficiency. Too often the purchaser is content with determining 
the first two (or even with no test at all if the machine runs and 
does some work). A test will require the service of a technical 
man, but under competent direction should not be difficult nor 
expensive. 

The head will be measured in inches of water in a U-tube (see 
precautions, Art. 23a). The quantity may be determined by 
measuring velocity and area. Where very great quantities are 
passing, the annamometer is the most convenient instrument for 
measuring velocity, but it should not be depended on in unskilled 
hands. It will need careful rating and should be applied in 
all parts of the cross-section of the conduit, and the total 
quantity found by summing the products of small areas by 
their respective velocities. In doing such work the operator's 
confidence in the method is apt to be shaken by the discovery 
that the velocity will vary considerably over the area and will 
also vary with time at a fixed point. In any case, the section 
at which the velocity is taken should be well away from the 
machine, else the irregular currents will render the observations 
altogether unreliable. 

The author is partial to orifice measurement, even for testing 
the largest fans. Orifice coefficients are now available up to 30 
in. diameter or 30 in. square (see Art. 23). It is the author's 
opinion that a coefficient of 0.60 will result in errors well within 
those made in reading water gages, and quite certainly with 
errors less than are apt to enter any annamometer or petot-tube 
measurements. Note that the orifice coefficient is constant, 
while that of a petot tube or a revolving annamometer must be 
found for each instrument and may change with the slighest 
injury or misuse of the instrument, and note further that with 
reasonable care that cross-currents are not allowed in front of the 
orifice, its discharge is not effected by unequal velocities in the 
cross-section of the conduit. 

Omitting any discussion of apparatus for measuring velocities, 
quantities and pressures, with their calibration and defects, the 
engineer will need to determine : 



102 



COMPRESSED AIR 



v = the velocity of air passing the section of area, a (feet per 
second), 
W = weight of air passing (pounds per second), 
P = pressure of air at section = 5.21 i where i is pressure in 

inches of water (pounds per square feet), 
N — revolutions per minute, u = 2rrN, 
Q = volume passing = av (cubic feet per second). 
Ei = power put into the fan (foot-pounds per second), 
E 2 = the useful work done by the fan. 
Then, 



and efficiency 



E% 
E~i 



Wv 2 



He should also have all dimensions and angles of the fan in 
order better to interpret results. 



D P< 













































I 


at 


12Q0 


R.P 


.M. 










' 






































$ 


&/ 


i 
i 


















— p 


-atJ 


nnri 




04 








< 






i 
i 
i 


6^. 


A 


















rt 






































§s 


oo 


























i 

i 


i 
i 


















Pi 


it 30 





u V 

\ \ 


U <U 






T"/* 






















JT^~ 




















V 


>l\ 




























^ x 




P a 


t COC 


// 




\J 




ii 


















\ 


— .. 










'/ 


/ 
































^ 


"^ 


A 


V. 


^ 


































































*^> 


^^ 








p at 4U0 

1 1 









3 4 5 G 7 

Cubic Feet per Minute in Thousands 

Fig. 22. 



The variables are N, v, and P. In a thorough test to get the 
performance chart of a fan, the preferable method is to maintain 
a constant N throughout a series of runs in which P is varied at 
will by the operator, v measured and E computed. 

Then, with another N another series is run as before and so 
covering the desired range for the fan. From these notes the 
performance curves can be drawn. The most important of these 
are those for efficiency and for quantity (see Fig. 22) . 



FANS 103 

The measures of v and P should be taken in a section of smooth 
straight conduit some distance from the fan. The pressure is 
controlled by placing some kind of shutter in the conduit beyond 
the section at which v and P are measured. 

A performance chart of the class shown in Fig. 22 shows in 
remarkable completeness what can, and should, be accomplished 
by the machine and under what conditions it will work most 
efficiently. On this chart the pressure curves and efficiency 
curves are plotted in the usual way as suggested above, then the 
efficiency contours are located thus. To locate the 55 per cent, 
contour, find the two points where the 1,000-r.p.m. efficiency 
line crosses the 55 per cent, line (at about the 5.2 and 7.7 ver- 
ticals). Shift these points vertically to the 1,000-r.p.m. pressure 
line and mark 55. Similarly find where the 800-r.p.m. efficiency 
line cuts the 55 per cent, efficiency line (at 3.6 and 6.). Shift 
these up (or down) to the 800-r.p.m. pressure line and mark 55 
as before, etc. Connect the points of equal efficiency by a 
curve. Similarly the 60 per cent, contour can be drawn. Then 
evidently the best combination for operating the machine is 
within the area surrounded by the 60 per cent, efficiency con- 
tour. For instance, if we want 7,000 cu. ft. per minute, the 
machine should be speeded to about 1,000 r.p.m. and at these 
rates the pressure would be about 7 in. Of if we want 5,000 
cu. ft. per minute the speed should be about 800 and the pressure 
about 5 in. 

Art. 53. Suggestions. — The following suggestions seem to be 
the rational conclusions pointed to by theory, the difficulties in 
controlling operation, and complications in analyzing the results 
of tests. 

Observing the rules as to smooth curves, polished surfaces, and 
correct angles, design a high-speed fan with characteristics as 
revealed in Fig. 23. DBE is an adjusting tongue hinged at D. 
By this area AB can be varied at will. The area of the sections 
gh, etc., are so proportioned as to maintain the velocity u in the 
volute until the throat, or switchpoint, A, is passed, after which 
the velocity is gradually reduced and pressure increased (by the 
well-known laws of fluid dynamics) in the trumpet-shaped outlet. 

In operation the intent would be to keep a constant pressure in 
the volute up to AB, this pressure head being approximately 

u 2 

sr- * Then the whole theory of this machine would be 



104 



COMPRESSED AIR 



_ _. u z , ' wau 6 

Q = au, H = — and E 2 = • 

9 g 

A factory test of such a machine would reveal the most favor- 
able relation between u and p. Then a size would be selected 
that would give the desired Q. In operation there would be a 



Adjuster Sliding Contact - 




Fig. 23. 

water gage tapped into the section AB to show p at any time. 
When the operator notes that p is low, he will open up the area at 
AB and vice versa. 

Note particularly that in such a design Q can be controlled 
independently of H. 



CHAPTER IX 
CENTRIFUGAL OR TURBO AIR COMPRESSORS 

Art. 54. Centrifugal Compression of an Elastic Fluid. — The 

demonstrations given in the preceding chapter apply to any case 
where there is no change of density of the fluid while passing 
through the machine, and this includes the case of centrifugal 
acceleration without change of pressure, and purely impulsive 
action. 

We have now to study the case where compression due to 
centrifugal force within the wheel, or runner, is so great that it 
must be considered in the formulas. 

We will assume isothermal conditions, since the ratio of com- 
pression in each stage is low and intercooling can be applied 
between each stage. The formulas thus gotten are simpler than 
can be gotten otherwise, and are as accurate as is justified by 
other considerations. 

In Fig. 19 assume the cylinder CB filled with a compressible 
fluid, as air. The weight of a unit volume will not be constant, 
but will depend on the distance x from the center and on the 
velocity of rotation. 

Let w x be the weight of a unit volume at distance x from the 
center. Then the centrifugal force due to a disk of unit area and 
radial thickness dx will be 



Also 



1 iV x^x 7 UJ %W 7 " **S 

ap x = dx = — r- xdx since u x = - u. 

gx gr z r 



Wi pi 



where p x is absolute pressure in the air at distance x from the 
center, and Wi and p x are the weight and pressure respectively 
of the air at entrance. 

Substituting and dividing by p x there results 



dj) x _ WiU 2 I 
Pi Px ViQT 2 Jri 



- L — = 5 xdx, then I - J — = 1 xdx. 

p x Pigr 2 Jpi p x Pigr- 

Whence 

105 



106 COMPRESSED AIR 

. p Wiu 2 /„ ,\ Wi/u 2 —Ui 2 \ . r 

log c — = — - — -lr 2 — ri 2 ) = — ( — = ) since n = -Mi. 

pi pi'2gr 2 \ I p x \ 2g I u 

If ri = and we consider a single-stage machine taking in free 
air we will have 

where R' is the ratio of compression at the periphery but within 
the revolving wheel. 

Assume that the machine is in 1 second putting a volume of free 
air = v a into the state of pressure and motion indicated above. 
Then the work done per second will be 

■1/2 9/ 2 

PaV a l0g c R' + W a V a 7T~ = W a^a ~ (&) 

^9 9 

when the value of log c R' from (a) is inserted. 

Note that this is the same as would result if the machine were 
working on an inelastic fluid of weight w a (see Art. 50). 

Note also that the work done in compression is equal to that 
done in giving velocity. 

Art. 55. A More Direct Derivation of Equation (37). — Appli- 
cable also to Centrifugal Air Compressors: 

After a study of Arts. 49, 50 and 54 the student will be pre- 
pared for the following more general and more direct demonstra- 
tion of Eq. (37) and its application in case of considerable com- 
pression. 

Referring to Figs. 24 and 20. The static pressure in the fluid 
changes as it passes out of the rotating part into the fixed out- 
let passage. It is this drop in pressure that induces the relative 
discharge velocity, V. This difference in pressure offers no 
resistance to the rotation of the wheel; as will be readily seen if 
we imagine the perifery of the wheel closed while rotating in a 
frictionless fluid. The pressure in the frictionless fluid must be 
normal to the perifery and therefore does not resist its rotation. 

Then in all cases (regardless of change of pressure at outlet) 
the resistance to rotation is due solely to the reaction of the 
departing jet. This reaction is in direction opposite to that of 
the absolute velocity of discharge, v, (Fig. 19) and in amount 

v 
is W-. But the component opposed to rotation (that is in direc- 

v 
tion opposite to u) is W~ cos 6 and as is apparent on the dia- 



CENTRIFUGAL OR TURBO AIR COMPRESSORS 107 

grams v cos 6 = u + V cos 8. Therefore the force opposed to 

W 
rotation is — (u -\- V cos 8) and since work done by the wheel 

equals force multiplied by distance. Then 

W 
Work = — (u 2 + uV cos 8). 

g 

Evidently this is independent of the radial depth (r — r x ) of the 
vanes. Then the radial depth of vanes is a matter of convenience 
or expediency. 

In case of a fluid of uniform density (in low pressure fans we 
may neglect change of density) if the machine imparts a head, 
H, to the fluid, then work = WH: Whence 

W 

WH = — (u 2 + uV cos 8) 
(J 

and 

= u 2 + uV cos 8 

g 

In case of a compressible fluid (as air) and we are to consider 
the work done in compression. 

Let Ri be the final ratio of compression when the air has been 
brought to rest after one stage. Then work = p a v a log c R: 
where v a is the volume of free air compressed. Then 



and 



y a Va log c #i = — — (u 2 + uV cos 8) 

w a (u 2 + uV cos 8\ 
log e Bi = —(- -— -) (37a) 

This is the formula for ratio of compression produced by one 
stage of a centrifugal air compressor. 

All the discussion in Art. 51 concerning the effects of the angle 
8 applies also to equation (37a). The student should read that 
article as a part of this study. 

If there are n stages in a machine, each giving an additional 
ratio, R h and the final ratio from free air be R n , then 

R n = R\ n and log c R n = n log c R\ (38) 

Art. 56. Working Formula. — The very great centrifugal force 
developed in these machines prompts manufacturers generally to 
prefer to set the outer tips of the propeller blades radial {8 = 90°) 



108 COMPRESSED AIR 

to avoid cross bending. This is good practice for other reasons 
(see Art. 51) one being that formulas for designing and analysis 
are much simplified, as the following will show: 
In Eq. (37a) assume p = 90°. Then 

. W a U 2 

l0&Ri = y«7 

7) W 

Note that w = ,. Q Q - . and, if t be assumed constant, — - will 

be constant. To adopt the formula to common logarithms (which 
will be more convenient) divided by 2.3026. 

In such a machine perfect cooling cannot be accomplished. 
We will assume for this study an average temperature of 580 
(120°F.). Then the formula becomes 

l0gl ° Rl = (2.3026 X 53.35X 580 X 3T2) U * = fcu ' (39) 
log k = 7.6398. 

In the following examples (taken from practice) j8 = 90°. 

Example 1. — A three-stage machine, 54 in. in diameter, r.p.m. 
2,500, in operation, gives 15 lb. gage pressure and delivers 35,000 
cu. ft. per minute of free air, the power necessary being 2,700 hp. 

Determine the efficiency of the machine as to pressure and 
power. 

Here u ^^ X 3.14 X ~ and logw 2 = 5.5368 u 590 

60 12 

log k = 7.6398 
log of log #x = T.1766 

logfli = 0.1500 #1 1.40 
3 

log R n = 0.4500 R n 2.83 

Assuming p a = 14.4 where the machine is in operation, 
then p = 2.83 X 14.4 = 40.75 and gage pressure = 40.75 — 
14.4 = 26.3. 

15 

Then efficiency as to gage pressure ^-5 = 57 per cent, and 

theoretic efficiency as to work would be, by Eq. 32, 
log R log 2.04 0.3096 



log R n log 2.83 0.4518 



68 per cent. 



The report of the test of the machine gave the "shaft" effi- 
ciency as 71 per cent., the meaning not being further defined. 



CENTRIFUGAL OR TURBO AIR COMPRESSORS 109 

Example 2. — A single-stage machine, 34 in. in diameter with 
3,450 r.p.m. gave 3.25 gage pressure and the horsepower was 350 
for 18,000 cu. ft. per minute. 

What efficiency as to pressure and power did the machine show? 

q 4 f^O 34 

lL = ^p X 3.14 X ^2 and logu 2 = 5.4048 

log k = 7.6395 
log of log Ri = T.0443 
logfli = 0.1107 
Ri = 1.29 

Assuming 14.5, then P = (1.29 - 1) 14.5 = 4.2 nearly. 

14.5 + 3.25 
The ratio efficiency would be — ... - J — = (1.22) divided by 

3.25 
1.29 = 95 per cent, and gage pressure efficiency = -j^ = 77 

per cent. 

Horsepower necessary to compress 18,000 cu. ft. per second 

to R = 1.22 is 218. Therefore, the efficiency as to power = 

218 

^~. = 63 per cent. 

Example 3. ■ — A six-stage machine 27 in. in diameter, r.p.m. 
3,450, gives 15 lb. gage and 340 hp., capacity 4,500 cu. ft. per 
minute. 

3 450 27 

u = -^- X 3.14 X j2 lo S w2 = 5 - 1976 

log k = 7.6395 
log of log R x = 2.8371 

logfli = 0.0687 Ri = 1.17 

6 

log R n = 0.4122 

R n = 2.58, P n = 37.5 
P g = 23 lb. 

The ratio accomplished by the machine is 2 very nearly; there- 

2 
fore, the ratio efficiency ^-^ = 77 per cent. 

The work necessary to compress 4,500 cu. ft. per second to 
R = 2 is 198. Therefore, the work efficiency is 58 per cent. 

When pressure is low, as in Ex. 2, the estimated efficiencies will 
be materially effected by the atmospheric pressure and the pres- 



110 



COMPRESSED AIR 



sure developed should be determined by a water or mercury 
column; otherwise only rough approximations will be obtained. 

Art. 57. Suggestions. — The following considerations point to 
the conclusion that best results will be gotten from centrifugal air 
compressors when the air is held in the machine until every par- 
ticle is under full centrifugal pressure regardless of its position 
relative to the propellers. Then it will escape through the outlet 
passages with uniform velocity and pressure, a condition evi- 
dently essential to high efficiency. Otherwise, if the air is still 
under the impulsive pressure of the vanes as it escapes from the 
machine, those particles next the propeller as at d, Fig. 24, must 
be under greater pressure than those at c, and the velocity of 




Fig. 24. 



escape relative to the revolving machine will be greater at d than 
at c. 

In these machines the velocity of rotation, u, is always very 
high and any moderate relative velocity of discharge (say within 
100 ft. per second) will leave the absolute path of the escaping air 
nearly on a tangent to the perimeter, as at ab. This being the 
case, a flaring fixed receiving passage about as shown at (6), 
Fig. 24, would cause the velocity to be gradually checked. It 
is not apparent that any advantage will be gotten by putting 
tongues ef in this outlet passage. They would increase friction 
without apparent compensating benefit. 

Note that a section of the flaring outlet on a horizontal plane 
through ab will show a much longer path than the radial section 
(see (c), Fig. 24). 

The very great centrifugal stress in these machines lead manu- 



CENTRIFUGAL OR TURBO AIR COMPRESSORS 111 

facturers generally to prefer to set the outer end of the propellers 
radial, and this is good practice for other reasons, one being the 
simplified formula for designing. 
Art. 58. Proportioning. — 

Let v a = volume of free air to be compressed, cubic feet 
per second, 
r = radius to outlet of propeller, 
ri = radius to inlet of propeller, 
u = velocity of rotation at outlet, 
Uy = velocity of rotation at inlet, 
S = radial component of the velocity at outlet, 
Si = radial component of velocity of air entering at 

radius ri, 
4> = angle between forward direction of U\, and tangent 

to vane at inlet, 
b = width of outlet, 
61 = width of inlet. 

All linear units in feet. 

R' = ratio of compression of air within the wheel at the outlet 

but before escaping, 
Ri = ratio of compression when brought to rest at end of first 

stage. 
Then 

tan d> == — 

Mi 

and V a = 2 wribiSi. 

Usually 61 is to be determined by this relation, all other factors 
being known. 

Note that this equation holds only when Si is the radial com- 
ponent of outward movement into the vanes. When there are 
no guide vanes at entrance, Si becomes uncertain and erratic. 
When it becomes the practice to put guide vanes at entrance, 
much of the uncertainties in the design of such machines will be 
removed. 

If there are to be n stages and a final ratio of compression 
1 
R n , then Ri = R n n and R' = Ri*. u will be fixed by the re- 
lation from (38) 



112 COMPRESSED AIR 

When u is determined, r and r 1 can be assigned between limits 
found advisable by experience and the necessity of having pas- 
sages of sufficient area. 

At the outlet the width b is fixed by the relation 

^- = 2irrbS. 

The greatest difficulty in the theoretic design of this class of 
machines is in correctly predicting the factor S (or relative ve- 
locity of discharge) . It is, of course, quite sensitive to changes of 
pressure in the discharge ducts. It is this doubtful factor 
chiefly that forces the designer to depend on results of tests. 
Fortunately, the width can be varied without affecting any 
other factors except V a . Hence, after test of a design of machine 
the desired capacity can be gotten by varying b and &i. 

The discharge of a centrifugal blower can be made adjust- 
able without varying the pressure by the simple device shown in 
Fig. 23. 

Finally, the student should be reminded that the above 
mathematical formulas do not include losses due to friction nor 
imperfections of design. Their chief value is to show what 
would be realized in a perfect machine and so reveal the short- 
comings of a machine and guide the designer in modification for 
improvements. 



CHAPTER X 
ROTARY BLOWERS 

Art. 59. — In certain lines of manufacture, there is necessary a 
supply of air in great volume, and at pressures not found prac- 
ticable for fans and yet so low, that to build reciprocating com- 
pressors to meet the demand would seem extravagant, when the 
cost is compared to the power demanded. 




Fig. 25. 

This demand has, in the past, been most economically met by 
the class of machines known as rotary blowers. These vary in 
details and there are several patterns on the market. Perhaps 
the simplest and best known is illustrated in Fig. 25. The two 
propellers revolve as shown by the arrows, and as is apparent by 
inspection the pockets of fluid (air or water) are forced upward. 
The flow is continuous, but not uniform ; neither is the tort on the 
shafts constant. The irregular discharge and tort tend to cause 
8 113 



114 COMPRESSED AIR 

vibrations but this is met by making the machines heavy and 
rigid (and in case of pumps by putting air chambers near both in- 
let and outlet). There are no valves in the machine, and the 
makers do not design them to rub or move in contact at any sur- 
face within the casing, but depend on accurate workmanship to 
make the clearance between the surfaces so small as to render the 
leakage so small as to be tolerable. 

Then, having no valves nor rubbing surfaces, the machines 
handling air should be quite durable, but this cannot be said of 
them, when used to pump water containing any grit. It is a good 
practice to supply a liberal quantity of thick oil to a blower, not 
for lubrication, but to reduce clearance between the surfaces. 

It is necessary to note one important difference in the working 
of this class of machine as a blower, and as a pump. This is due 
to the reexpansion of the compressed air out of chamber B into A, 
as soon as communication is opened between the two chambers. 
This is lost work and would limit the pressure at which the ma- 
chine could operate economically, even if slippage did not increase 
with pressure. For these reasons the useful range of pressures 
on such machines seems to be between }4 and 5 lb. For pres- 
sures below }4 lb. fans are usually selected, on account of the less 
cost. For pressures above 5 lb. reciprocating compressors are 
usually selected, on account of the better efficiency. 

Apropos to this phase of the subject, read Chapter IX on 
"Centrifugal Air Compressors." 



CHAPTER XI 
EXAMPLES AND EXERCISES 

Art. 60. — The following combined example includes a solution 
of many of the types of problems that arise in designing com- 
pressed-air plants. The student will find it well worth while to 
become familiar with every step and detail of the solutions, which 
are given more fully than would be necessary except for a first 
exercise. 

Example 60. — An air-compressor plant is to be installed to 
operate a mine pump under the following specifications : 

1. Volume of water = 1,500 gal. per minute. 

2. Net water lift = 430 ft. 

3. Length of water pipe = 1,280 ft. 

4. Diameter of water pipe = 10 in. 

5. Length of air pipe = 1,160 ft. 

6. Atmospheric pressure = 14 lb. per square inch. 

7. Atmospheric temperature 50°F. 

8. Loss in transmission through air line = 8 per cent, of the 
pv log e r at compressor. 

9. Mechanical efficiency of the pump = 90 per cent, as re- 
vealed by the indicators on the air end and the known work 
delivered to the water. 

10. Average piston speed of pump = 200 ft. per minute. 

11. Mechanical efficiency of the air compressor = 85 per cent, 
as revealed by the indicator cards. 

12. Revolutions per minute of air compressor = 90 and vol- 
umetric efficiency = 82 per cent. 

13. In compression and expansion n = 1.25. 

Preliminary to the study of the problems involving the air we 
must determine: 

(a) Total pressure head against which the pump must work. By 
the methods taught in hydraulics the friction head in a pipe 10 
in. in diameter, 1,280 ft. long, delivering 1,500 gal. per minute, is 
about 20 ft. Therefore, the total head = 450 ft. 

115 



116 COMPRESSED AIR 

(b) Total work (Wi) delivered to the water in 1 min. 
Wi = 1,500 X SM X 450 = 5,625,000 ft.-lb. 

(c) Total work (W) required in air end of pump. By specifica- 
tion 9, W = ^ = 6,250,000 ft.-lb. = 190 hp. 

For the purpose of comparison, two air plants will be designed; 
the first, designated (d) as follows: 

(d) Compression single-stage to 80 lb. gage. No reheating. 
No expansion in air end of pump. Pump direct-acting without 
flywheels. 

Determine the following: 

(dl) Air pressure at pump and pressure lost in air pipe. By 
specification 8 and Eq. (32), 

100 " — 80 + 14' ° r l0g 14 = 0!)2 l0g 6 ' 72 - 

log ™T4— 

P 2 

Whence, using common logs, log y^ = 0.76118 and 

p 2 = 80.78. 

Then lost pressure = p x - p 2 = 94 - 80.78 = 13.22 = f, 
and gage pressure at pump = 80 — 13.22 = 66.78. 

(d2) Ratio between areas of air and water cylinders in pump. 
The pressure due to 450 ft. head = 450 X 0.434 = 194.3, say 
195 lb., per square inch; and since pressure by area must be 

■ , . area air end 195 _ . 

equal on the two ends, r -j — nn no = 3 nearly. 

M area water end 66.78 

(d3) Volume of compressed air used in the pump. Cubic feet 
per minute. Evidently from solution (d2) the volume of com- 
pressed, air used in the pump will be three times that of the 
water pumped, or 

v = I t c X 3 = 601.6 cu. ft. per minute. 

7.48 

(d4) Diameters of air cylinder and of water cylinder. Since the 
piston speed is limited to 200 ft. per minute (specification 10) and 
the volume is 1,500 gal., we have, when all is reduced to inch units 
and letting a = area of water cylinder, a X 200 X 12 = 1,500 
X 231. Whence a = 144 sq. in. which requires a diameter of 
about 13% in. 



EXAMPLES AND EXERCISES 117 

The area of air cylinder is by {62) three times that of the water 
cylinder, which gives a diameter 23)^ in. for the air cylinder. 

{65) Volume of free air. From {61) r at the pump = 5.76. 
Therefore 

v a = 601.6 X 5.76 = 3,465 cu. ft. per minute. 

{6Q) Diameter of %ir pipe. The mean r in the air pipe is 

- — -^— = 6.24. Using this in Eq. (27) with c = 0.06, we 

get 6 = 5 in. 

13 22 
Or using Plate III with r X 13.22 -f- 1.160 or r X yJqq 

on the fr line and 3,465 on the V a line, the intersection falls near 
the 5-in. point on the d line. 

{67) Horsepower require6 in steam en6 of compressor. By 
Table II the weight per foot of free air is 0.07422 lb. per cubic foot. 
Total weight of air compressed = Q. 

Q = 0.07422 X 3,465 = 257 lb. per minute. 

In Table I opposite r = 6.72 in column 9 find by interpolation 
0.3736. Then 

Horsepower = 2.57 X 0.3736 X (460 + 50) = 489.6 in air 

end, and tt^V - = 576 in steam end. 
0.85 

The second plant will be designated by the letter (e) and will be 
two-stage compression to 200 lb. gage at air compressor, will be 
reheated to 300° at the pump and used expansively in the pump; 
the expansion to be such that the temperature will be 32° at end 
of stroke. 

(el) Air pressure at pump. Apply Eq. (32) as in {61). In this 
case r\ (at the compressor) == 15.3 and r 2 (at the pump) = 12.3. 
Therefore pressure at the pump = 12.3 X 14 = 172.3 and the 
lost pressure = 214 - 172.3 = 41.7 = f. 

(e2) Point of cutoff in air en6 of pump = fraction of stroke 

6uring which air is a6mitte6. By Eq. (12), viz., — = (— ) , 

t\ \Vy 

492 /vA ° -25 Vi 

putting in numbers we get =ttx = ( — ) whence — = 0.176, which 

^ ° 760 \vil v 2 

is the point of cutoff, and v 2 = 5.68 V\. 

Or go into Table I in column 5, find the ratio ^qo = 1-545, 
and in same horizontal line in column 3 find 0.176. 



118 COMPRESSED AIR 

(e3) Volume of compressed hot air admitted to air end of pump. 
Apply Eq. (9), viz., work = — 1 f- piVi - p a v 2 . 

In this we have work = 6,250,00, v 2 = 5.68 Vi, p\ = 214, 
n — 1 = 0.25, p a = 14, and p 2 must be found by Eq. (12a), or 
it may be gotten from Table I by noting that for a tempera- 
ture ratio of 1.545 the pressure ratio is 8.8 and - = 0.1136. 

Therefore p 2 = 0.1136 X 172.3 = 19.57. This would give gage 
pressure = 5.57. 

Inserting these numerals in Eq. (9) we get 

6,250,000 = 144^( 172 - 3 ~^ X 19 ' 57 + 172.3 - 14 X 5.68) . 

Whence v\ = 128.6 cu. ft. per minute. 

(e4) Diameter of air cylinder of pump when air and water 
pistons are direct-connected. Since expansion ratio is 5.68 (see 
(e2)) and the volume before cutoff is 128.6, the total piston dis- 
placement is 128.6 X 5.68 = 730.8 cu. ft. per minute. When 
the air and water pistons are direct-connected they must 
travel through equal distances, therefore, the air piston travels 
through 200 ft. per minute (specification 10). Then if a = area 
of piston in square feet we have 

200 a = 730.8 and a = 3.654 sq.ft. 

By Table X the diameter is 26 in. nearly. 

(e5) Volume of cool compressed air used by pump, cubic feet 
per minute. By (e3) the volume of hot compressed air is 128.6, 
and since under constant pressure volumes are proportional to 
absolute temperatures, we have 

v 510 



Whence v = 86.3 cu. ft. per minute. 



128.6 760" 

(e6) Volume of free air used. From (el) the ratio of compres- 
sion at the pump is 12.3 and from (e5) the volume of cool com- 
pressed air is 86.3, therefore, the volume of free air is 86.3 X 12.3 
= 1,061.6. 

(e7) Diameter of air pipe. The r for Eq. (27) is 

12 - 3 + 153 = 13.8. 



EXAMPLES AND EXERCISES 119 

Applying Eq. (21) with coefficient c = 0.07 we have 

1,061. 6\V* 



0.07 X 1,160 x (i5"±) 



d = \~ 41.7 X 13.8 / = 2 - 13in - 

(e8) Horsepower required in steam end of compressor. By (d7) 
the weight per cubic foot of free air is 0.07422 and by (e6) the 
volume of free air compressed is 1,061.6 Therefore, the total 
weight compressed is 0.07422 X 1,061.6 = 78.8 lb. per minute, 
and the initial absolute temperature is 510. 

In the two-stage compression r 2 = 15.3, and assuming equal 
work in the two stages the r\ = \/l5.3 = 3.91 nearly (see 
Art. 13). Then going into Table I with r = 3.91 in column 9 find 
0.2525. Hence horsepower = 0.2525 X 78.8 X 510 = 101.5 for 
one stage, and for the two stages 101.5 X 2 = 203, 

203 
and (specification 11) w~Qk = 238.8 hp. in steam end. 

(e9) Diameter of air compressor cylinders, assuming 3-ft. strokes 
and 2%-in. piston rods, equal work in the two cylinders and allowing 
for volumetric efficiency. By (e6) the free air volume is 1,061.6 
and (specification 12) the volumetric efficiency = 82 per cent. 
Therefore, 

the piston displacement = ' ' = 1,294.6 cu. ft. per minute. 

By specification 12 the r.p.m. = 90. Therefore, the displace- 
ment per revolution = 14.7 nearly, for the low-pressure cylinder. 
Add to this the volume of one piston rod length of 3 ft. which is 
3 X 0.341 = 0.1023. Whence the volume per revolution must 

7.4 
be 14.8 or, for one stroke, 7.4. Whence the area = -tt = 

2.466 sq. ft. By Table X the diameter is 213^ in. nearly for low- 
pressure cylinders. 

The high-pressure cylinder must take in the net volume of air 
compressed to r = 3.91 (see (e8)). Therefore, the net volume per 

revolution = Q J ^ Q1 = 3.02. Add one piston rod volume 

and get 3.12 per revolution or 1.56 per stroke and an area of 0.53 
sq. ft. By Table X this requires a diameter of 10 in. nearly. 

(elO) Temperature of air at end of each compression stroke. In 
Table I the ratio of temperatures for r = 3.91 is 1.313. Hence the 
higher temperature = 510 X 1.313 = 669 absolute = 209°F. 



120 



COMPRESSED AIR 



DESIGN OF A SYSTEM OF DISPLACEMENT PUMPS 

The water in a mine is to be collected by a system of displace- 
ment pumps, one each at B, C, D and E, delivering into a sump 
at A. 

The data are shown on the sketch (Fig. 26) and include: 
lengths of pipes (I) ; elevations (El) and quantities (Q) of water 
in cubic feet per minute. 

The lengths of water pipes and air pipes will be assumed equal. 
The pipes may change diameter at junctions. Assume one-third 
of time consumed in filling the tanks with water. Then the 
maximum rate of discharge must be three-halves of the average. 

The problem is to specify the free air volume (V a ) for the com- 
pressor and the gage pressure (P) of delivery. Also, the diame- 
ters of all pipes both for water and for air. 



1=800- 



^ 



-1=800- 



-1 = 900- 



El=60 



■1=1400 



El =45 
Q = 12 



t>U 



Qj El=40 
B Q= io 



El=35 



El =30 



EH 



Q =20 e 



Fig. 26. 



Solution. — In order to avoid putting in reducer valves and for 
economy in piping we will as nearly as practicable, design the 
system so that static head + friction head in air pipe + fric- 
tion head in water pipe shall be the same for each unit. Evi- 
dently this sum will be fixed by conditions at E, since it has 
greatest lift and greatest length of pipe. 

We will, therefore, first fix diameters for lines EH and HA, 
giving them liberal dimensions in order to keep down the pressure 
at A for we will find that some of the pressure at A must be 
wasted when working the pumps at B, C and D. 

The following computations were made with the aid of slide 
rule and logarithmic friction charts (Plate III), such method 
being sufficiently accurate for the purpose. 



EXAMPLES AND EXERCISES 



121 



Water line EH : 
6 in. diameter 

Water line HA : 
6 in. diameter 



Air pipe EH: 
2 in. diameter 

Air pipe HA : 
2 in. diameter 



[ Length 1,400ft., Q = ^ X 20 =30 
Pressure loss in 1,400 ft. 
(pounds per square inch = 4.0 

Length 800, Q =42 

Pressure loss in 800 ft. = 3.8 

Water friction E to A = 7.8 

Static pressure E to A = 13.0 

Pressure on water at E = 20.8 

For 20.8 lb. gage r = 2.44, but at A the air 
pressure must be somewhat greater. Hence, 
we may assume r = 2.5 for estimating friction 
in pipes leading from A. 
Length 1,400 ft. volume of compressed air 
V c = 30 

V a = r X 30 = 75, air friction E to H,f = 2.3 

Length = 800 ft. V c = 42, V a = 105, / = 2^4 

Air friction E to A = 4.7 

Air pressure at A = 20.8 + 4.7 = 25.5 

r at A = 2.78 



Air pipe A to Compressor 
2 in. diameter 



;i2 + 



Length 500 ft., V c = 
10 + 8 + 20) = 75 
V a = r X 75 = 210, /= 4.8 

At compressor P = 30.3 
The assumption that all pumps 
will discharge simultaneously is 
extreme. Hence a compressor of 
200 cu. ft. per minute and gage 
pressure = 30 lb. will be ample. 
Now with air pressure = 25.5 at junc- 
tion A , we have to design the air and water 
pipes to pumps B, C and D so as to about 
use up this pressure. 
Air pipe GA: ) Length 800 ft., V c = 33, V a = 2.5 V c , 
1% in. diameter J V a = 82, / = 3.2 

Air pipe BG: \ L th 200 ft y = 15 y = 37 j = 2A 
1 m. diameter J 

Air pipe CG: 1 th = = 45, / = 5.4 

1>4 in. diameter J 

From the above we find air pressure in 
tank B = 25.5 - (3.2 + 2.4) = 18.9 

tank C = 25.5 - (3.2 + 5.4) = 16.9 



122 COMPRESSED AIR 

Water pipe AG : J Length = 800 ft., Q = 33, friction loss 
5 in. diameter J (pounds) = 6.1 

Static pressure at B (20 ft.) = 8.7 and 
18.9 - (8.7 + 6.1) = 4.1 

for water friction BG 
Water pipes BG: Length = 200 ft., Q = 15, loss of pres- 

Z}i in. diameter sure = 2.2 

Leaving a margin of 1.9 lb. 

Water pipe CG : Length 800 ft., Q = 18, static pressure 

4 in. diameter (15 ft.) = 6.5 and 16.9 - 6.5 = 10.4 that 
can be lost in friction in the water pipe. 
A 4-in. pipe will take up 6.2 leaving a 
margin of about 4 lb. This is the nearest 
commercial size that can be used. 

Air pipe DH : Length = 100 ft., V c = 12, V a = 2.5 X 

Y± in. diameter 12 = 30, / = 3.6 

Air pressure in D = 25.5 — air friction in 
AH and HD = 25.5 - (2.4 + 3.6) = 19.5 
Static water pressure at D (25 ft.) = 10.9 

Available for water friction = 8.6 

Water pipe DH: Length 100 ft., Q = 12, loss in 2^-in. 

2}i in. diameter pipe = 5.9 

Nearest commercial size, Margin 2.7 

EXERCISES 

In the following exercises, where not otherwise specified, atmospheric con- 
ditions may be taken as T = 60°F. and p a = 14.7. 

The article of the text on which the solution chiefly depends is indicated 
thus ( ) and the answer thus [ ]. 

1. (a) Assuming isothermal conditions, how many revolutions of a com- 
pressor 16-in. stroke, 14-in. diameter, double-acting, would bring the pressure 
up to 100 lb. gage in a tank 4 ft. diameter by 12 ft. length, atmospheric pres- 
sure = 14.5 per square inch? (1) [361]. 

(b) What would be the horsepower of such a compressor running at 100 
r.p.m.? (1) [37.3]. 

(c) What would be the horsepower if the compression were adiabatic? 
(2) [51.0]. 

(d) What weight of air would be passed per minute when r.p.m. = 100 and 
T = 60°F.? (8) [21.4]. 

2. The air end of a pump (operated by compressed air) is 20 in. in diameter 
by 30-in. stroke, r.p.m. = 50, cutoff at Y± stroke, free air pressure = 14.0, 
T a = 60°, compressed air delivered at 75 lb. gage, T = 60° and n = 1.41. 

(a) Find work done in horsepower. (3) [70]. 



EXAMPLES AND EXERCISES 123 

(6) Find weight handled per minute. (8) [56]. 

(c) Find temperature of exhaust (degrees F.). (7) [ — 165]. 

3. With atmospheric pressure, p a = 14.7, and T a = 50°, under perfect 
adiabatic compression, what would be the pressure (gage) and temperature 
(F.) when air is compressed to: 

(a) % its original volume? (7) [210]. 

(6) % its original volume? -(7) [435]. 

(c) }i its original volume? (7) [603]. 

(d) H its original volume? (7) [737]. 

(e) Ko its original volume? (7) [852]. 

4. With Pa = 14.1 and T a = 60° what will be the pressure of a pound of 
air when its volume = 3 cu. ft.? (8) [51.4]. 

5. What would be the theoretic horsepower to compress 10 lb. of air per 
minute from p a = 14.3 and T a = 60° to 90 lb. gage? 

(a) Compression isothermal. (1) [16.7]. 

(b) Compression adiabatic. (2) [22.7]. 

6. Find the point of cutoff when air is admitted to a motor at 250°F. and 
expanded adiabatically until the temperature falls to 32°F. (7) [0.41]. 

7. What is the weight of 1 cu. ft. of air when p„= 14.0 and T a = - 10°? 
(8) [0.84]. 

8. A compressor cylinder is 20 in. in diameter by 26-in. stroke double- 
acting. Clearance = 0.8 per cent., piston rod = 2 in., r.p.m. = 100, 
atmospheric pressure, p a = 14.3, atmospheric temperature = T a = 60°F., 
and gage pressure = 98 lb. 

Determine the following: 

(a) Compression isothermal. 

la. Volume of free air compressed, cubic feet per minute. (46) [891]. 
2a. Volume of compressed air, cubic feet per minute. (1) [1,144]. 
3a. Work of compression, foot-pounds per minute. (1) [3,757,000]. 
4a. Pounds of cooling water, Ti = 50°, T 2 = 75°. (9) [193]. 

(b) n = 1.25 and air heated to 100° while entering. 

lb. Volume of free air compressed per minute. (46) [830]. 
2b. Volume of cool compressed air per minute. (1) [106.5]. 
36. Work done in compression. (1) [4,658,000]. 
46. Temperature of air at discharge. (7) [385°F.]. 

9. The cylinder of a compressed-air motor is 18 by 24 in., the r.p.m. = 90, 
air pressure 100 lb. gage. In the motor the air is expanded to four times its 
original volume (cutoff at %), with n = 1.25. 

(a) Determine the horsepower and final temperature when initial T = 
60°F. (3 and 7) [hp. = 132, T = -90]. 

(6) Determine the horsepower and final temperature when initial T = 
212°F. (3 and 7) [hp. = 132, T = +17]. 

10. Observations on an air" compressor show the intake temperature to 
be 60°F., the r = 7 and the discharge temperature = 300°F. What is the 
n during compression? 

Hint.— Use Eq. (11a) with n unknown. (7) [1.25]. 

11. In a compressed-air motor what percentage of power will be gained 
by heating the air before admission from 60° to 300°F.? (2) [46 per cent.]. 



124 



COMPRESSED AIR 



12. If air is delivered into a motor at 60°F. and the exhaust temperature 
is not to fall below 32 °F., what ratio of expansion can be allowed? What 
could be allowed if initial temperature were 300°? n = 1.25. (2 and 7) 
[1.31, 8.8]. 

13. A compressed-air locomotive system is estimated to require 4,000 
cu. ft. per minute of free air compressed to 500 lb. gage in three stages with 
complete cooling between stages. 

Assume n = 1.25, p a = 14.5, T a = 60°, vol. eff. = 80 per cent., mech. 
eff. = 85 per cent, and r.p.m. = 60. 

Compute the volume of piston stroke in each of the three cylinders and the 
total horsepower required of the steam end. (13 and 14) [41.5, 12.7, 3.87, 
1,220]. 

14. A compressor is guaranteed to deliver 4 cu. ft. of free air per revolu- 
tion at a pressure of 116 (absolute). To test this the compressor is caused to 
deliver into a closed system consisting of a receiver, a pipe line and a tank. 
Observed conditions are as follows: 



Receiver 



Pipe 



Tank 



Pressure at start (ab.) . . . 
Temperatures at start (F. 
Pressures at end (ab.) . . . 
Temperatures at end (F.) 
Volumes (cubic feet) .... 



14.5 

60.0 

116.0 

150.0 

50.0 



14.5 
60.0 
116.0 
90.0 
10.0 



14.5 
60.0 

116.0 
60.0 

100.0 



How many revolutions of the compressor should produce this effect? 
(27) [264]. 

15. Find the discharge in pounds per minute through a standard orifice 
when d = 2 in., i = 5 in., t = 600° and p a = 14.0. (21) [8.03]. 

16. What diameter of orifice should be supplied to test the delivery of a 
compressor that is guaranteed to deliver 1,000 cu. ft. per minute of free air? 
(21) [6.5]. 

17. What is the efficiency of transmission when air pressure drops from 
100 to 90 lb. (gage) in passing through a pipe system? (31) [95.5]. 

18. A compressor must deliver 100 cu. ft. per minute of compressed air 
at a pressure = 90 lb. gage, at the terminus of a pipe 3,000 ft. long and 3 in. 
in diameter. p a = 14.4, T a = 60°F. 

(a) Assuming a vol. eff. = 75 per cent., what must be the piston displace- 
ment of the compressor? [967]. 

(b) What pressure is lost in transmission? (29) [17]. 

(c) What horsepower is necessary in steam end of compressor if n = 1.25 
and the mech. eff. = 85 per cent.? (29 and 2) [141]. 

(d) What would be the efficiency of the whole system if air is applied in 
the motor without expansion, the efficiency to be reckoned from steam 
engine to work done in motor? (6) [27 per cent.]. 

19. It is proposed to convey compressed air into a mine a distance of 
5,000 ft. The question arises: Which is better, a 3-in. or a 4-in. pipe? 

Compare the propositions financially, using the following data: Nominal 



EXAMPLES AND EXERCISES 125 

capacity of the plant = 1,000 cu. ft. free air per minute, vol. eff. of com- 
pressor = 80 per cent., n = 1.25 gage pressure at compressor = 100, weight 
of free air w a = 0.074, p a = 14.36, weight of 3-in. pipe = 7.5 and of 4-in. 
pipe = 10.7 lb. per foot. Cost of pipe in place = 4 cts. per pound. Cost of 
1 hp. in form of pv log r for 10 hr. per day for 1 year = $150. Plant runs 
24 hr. per day. Rate of interest = 6 per cent. (29) [Economy of 4-in. 
pipe capitalized = $86,260]. 

20. Air enters a 4-in. pipe with 60 ft. velocity and 80 lb. gage pressure; 
the air pipe is 1,500 ft. long. 

(a) Find the efficiency of transmission. (31) [91 per cent.]. 

(b) Find horsepower delivered at end of pipe in form pv log r. (31) [224]. 

(c) Find horsepower delivered at end of pipe in form P g X v. (31) [73.5]. 

21. An air pipe is to be 2,000 ft. long and must deliver 50 hp. at the end 
with a loss of 5 per cent, of the pv log r as measured at compressor. The 
pressure at compressor is 75 lb. gage. p a = 14.7. Find diameter of pipe. 
(29) [2%]. _ 

22. Modify 21 to read: 50 hp. . . with loss of 5 per cent, of the energy 
in form P g X v, where P g is gage pressure, and find diameter of air pipe. 
(29) [SHI 

23. In case 21 let pressure at compressor be 250 lb. gage and find diameter 
of air pipe. (29) [1.4]. 

24. The air cylinder of a compressed-air pump is 20 in. in diameter by 30- 
in. stroke. The machine is double-acting and makes 50 r.p.m. The cutoff 
is to be so adjusted that the temperature of exhaust shall be 30°. p a = 
14.5 and the r at pump =8. n = 1.25. 

(a) Find cutoff when initial temperature is 60°F. [0.78]. 

(b) Find cutoff when initial temperature is 250°F. [0.226]. 

(c) Find horsepower in case (a). [223]. 

(d) Find horsepower in case (b). [112]. 

(e) In case (a) find efficiency in applying the pv log r of cool air. [55 
per cent.]. 

if) In case (b) find efficiency in applying the pv log r of cool air. [85 per 
cent.]. 

(g) Find the volumes of free air used in cases (a) and (b). [3,400 and 
732]. 

25. A compound mine pump is to receive air at 150 lb. gage; this is to be 
reheated from 60° to 250°F., let into the H.P. cylinder of the pump and ex- 
panded until the temperature is 32°, then exhausted into an interheater 
where the temperature is again brought to 250°. It then goes into the L.P. 
cylinder and is expanded down to atmospheric pressure = 14.5 (ab.). 

(a) Find point of cutoff in each cylinder, n — 1.25. [0.23 and 0.61]. 

(b) If the air is compressed in two stages with n = 1.25, what will be the 
efficiency of the system, neglecting friction losses? [1.06]. 

(c) How much free air will be required to operate the pump if it is to 
deliver 250 hp., assuming the efficiency of the pump to be 80 per cent, 
reckoned from the work in the air end? [1,686]. 

(d) If the pump strokes be 60 per minute and 60 in. long, fix diameters of 
air cylinders in case (c). [23 in. and 35 in.] 

26. Compute the horsepower of a motor passing 1 lb. of air per minute 



126 COMPRESSED AIR 

admitted at 200°F. and 116 lb. (ab.) r = 8, the air to be expanded until 
pressure drops to 29 lb. (ab.), r = 2. n = 1.25. (3 and 7) [1.727]. 

27. A pump to be operated by compressed air must deliver 1,000 gal. of 
water per minute against a net head of 200 ft. through 800 ft. of 10-in. pipe. 
The pump is double-acting, 30-in. stroke, 50 strokes per minute. The air is 
reheated to 275°F. before entering the pump. The cutoff is so adjusted that 
with n = 1.25 the temperature at exhaust = 36°F. . Mec. eff. of pump = 80 
per cent. Air pressure at compressor = 80 lb. gage, p a = 14.4. Length of 
air pipe = 2,000 ft. Permissible loss in transmission = 7 per cent, of the 
pv log r at compressor. Mec. eff. of compressor = 85 per cent. Vol. eff. 
= 80 per cent. 

(a) Proportion the cylinders of the pump. [Water 14 in., air 26 in.]. 

(b) Determine the volume of free air used. [444]. 

(c) Determine the diameter of air pipe. [3^]. 

28. Compare the volume displacement of two air compressors, one at 
sea level and the other at 12,000 ft. elevation; the compressors to handle 
the same weight of air. [9.45 -5- 14.7]. 

29. (a) An exhaust pump has an effective displacement of 3 cu. ft. per 
revolution. How many revolutions will reduce the pressure in a gas tank 
from 30 to 5 lb. absolute, volume of tank = 400 cu. ft.? (15) [239]. 

(b) If the pump is delivering the gas under a constant pressure of 30 lb. 
(ab.) what is the maximum rate of work done by the pump — foot-pounds 
per revolution? n = 1.25. (15) [5,433], 

30. An air-lift pump is to be designed to elevate gravel from a submerged 
bed. Specifications as follows: 

Depth of submergence = 50 ft. ; lift above water surface = 10 ft.; volume 
lifted to be y± gravel and % sea water; specific gravity of gravel = 3; weight 
of sea water = 65 lb. per cubic foot; volume of gravel = 1 cu. yd. per minute. 

(a) Determine the ratio ~q>Q = volume of mixed water and gravel. 

(b) Determine the ratio of compression and horsepower of compressor. 

(c) Recommend diameters for water pipe and for air pipe. (41). 



TABLES 



Notes on Table i 

The table is designed to reduce the labor of solution of formulas 12, 120, Sd 
and 1a. 

When the weight of air passed and its initial temperature are known, the 
table covers all conditions such as elevation above sea level, reheating and com- 
pounding, but it does not include the effect of friction and clearance. 

In compound compression the same weight goes through each cylinder. Then 
knowing the initial t and the r for each cylinder, find from the table the work done 
in each cylinder and add. Usually the r and / are assumed the same in each 
cylinder. In that case take out the work for one stage and multiply by the 
number of stages. 

The columns headed "Work Factor" are applicable in cases of expansion, 
only when the expansion is complete, that is, when final pressure in the cylinder 
is equal that outside. (In free air or in a receiver.) 

Example. — Air is received at such a pressure that r = 8. What should be the 
cutoff in order that the temperature drop from 60° to 32°F. when expansion is 
adiabatic? 

The ratio of absolute temperatures is 1.057 which by linea interpolation corre- 
sponds to a volume ratio 0.871 or cutoff is at %. 

What would be the pressure at exhaust? 

The two ratios above are in the horizontal line with - = .825 therefore the 

r 

final pressure = .825 X initial pressure. 

To find the foot-pounds per pound of air, multiply the number opposite r in 
columns 7, 8 or 11 as the case may be by the absolute lower temperature. 

To find the weight compressed, go into Table II with known atmospheric con- 
ditions and cubic feet capacity of the machine. 

To find the horse-power per 100 of air per minute multiply the number oppo- 
site r in columns 9, 10 or 12, as the case may be, by the absolute lower temperature. 



128 



Table L- 



-General Table Relating to Air Compression 
and Expansion 







Ratio of 


Ratio of 




Work Factor. 


Work Factor 
for Isothermal 


(3 


$"0 
K 


Less to 
Greater 

Volume — 


Greater to 
Less Tem- 
perature — 


Air Heated by Compression 


Compression 













u 2 


OO 


Tempera- 


Tempera- 


K 


n 


H.P. Fac- 




2 » 


p.' is 


o.fci 


tures 


tures Ab- 




tor per 100 


K 


<u 3 


ti d 


*"< 


Changing 


solute 




/ n \ 


Pounds per 


tH ^ O 
£J4J (0 


p,S 


O p, 

Is 


" 1 

CD 

a 
Co 


1 
Vl ( I \ n 
vz \ r ) 


,w — 1 

r-0" 


Factor K for one 


Minute 
K 


o*^ 

■»J Vl 

O CD 

rt p, 


4-> 

a 
Pi 


Pi 


\ 1 




pound 


330 




*5 
























n = 


n = 


n = 


n — 








n = 


n = 


K 






1.25 


1. 41 


I. 25 


1. 41 


n = 


1-25 


n = 1 .41 


1-25 


r.14 




330 


r 


1 


V2 
Vl 


V2 
Vl 




h 
h 


Ft. 


-Lbs. 


Ft.-Lbs. 


H.P. 
9 


H.P. 


Ft.-Lbs. 


H.P. 


i 


2 


3 


4 


5 


6 


7 


8 


10 


11 


12 


i 


1 . 0000 


1. 000 


I. OOO 


1 . 000 


1 .000 


O 





0.0 


.0 


.0 


0.0 


.0 


1. 1 


.9091 


.927 




935 


1. 019 


1.028 


5 


131 


5.140 


•OI55 


•OI55 


5.068 


•oi53 


1.2 


• & 333 


.862 




877 


1.037 


1.054 


9 


863 


9-932 


.0298 


.0301 


9.694 


.0293 


i-3 


.7692 


.812 




830 


1.054 


1.079 


14 


329 


14-45° 


•0434 


•0437 


I3-950 


.0422 


1.4 


•7i43 


.764 




787 


1.070 


1-103 


18 


5°3 


18.766 


.0560 


.0568 


17.890 


•0542 


i-5 


.6667 


•7 2 3 




75° 


1.085 


1-125 


22 


465 


22.827 


.0680 


.0691 


21-559 


•0653 


i.6 


.6250 


.687 




717 


1. 100 


1. 146 


26 


186 


26.704 


•0793 


.0809 


24.991 


•o757 


i-7 


.5882 


•654 




686 


1. 112 


1. 166 


29 


775 


3 -4i7 


.0902 


.0921 


28.214 


•0855 


i.8 


•5555 


.625 




659 


1-125 


1. 186 


33 


178 


33-985 


.1005 


.1029 


31-252 


•0947 


1.9 


•5 26 3 


•598 




634 


1 -137 


1.205 


36 


421 


37.422 


.1104 


•I 134 


34.127 


.1034 


2.0 


.5000 


•574 




612 


1. 149 


1.223 


39 


53° 


40.733 


.1198 


•1235 


36.855 


.1117 


2. 1 


.4762 


•552 




59° 


1. 160 


I.240 


42 


536 


43-897 


.1289 


•1330 


39-450 


.1196 


2.2 


•4545 


•532 




57i 


1. 171 


1.259 


45 


407 


46.988 


.1376 


.1424 


41.912 


.1270 


2 -3 


.4348 


.514 




553 


1. 181 


i ; 273 


48 


199 


49.970 


.1461 


•1514 


44-287 


•1342 


2.4 


.4166 


.496 




537 


1. 191 


1.289 


5° 


884 


52.878 


.1542 


.1602 


46.548 


.1411 


2 -5 


.4000 


.480 




522 


1.202 


1.304 


53 


462 


55-676 


. 1620 


.1687 


48.720 


.1476 


2.6 


.3846 


.466 




508 


1. 211 


I-3I9 


55 


988 


58.402 


.1697 


.1769 


50.805 


•1539 


2.7 


•37°4 


• 452 




493 


1.220 


1-334 


58 


434 


61.054 


.1771 


.1850 


52.811 


. 1600 


2.8 


•3571 


•439 




481 


1.229 


1.348 


60 


800 


63-651 


.1843 


.1929 


54-745 


.1659 


2.9 


.3448 


•427 




469 


1.237 


1.362 


63 


086 


66.175 


.1912 


.2006 


56.612 


•1715 


3-° 


•3333 


.415 




458 


1.246 


i-375 


65 


3J9 


68.626 


.1979 


.2080 


58.414 


.1770 


3-i 


.3226 


.405 




448 


1.254 


1.388 


67 


499 


71.158 


.2045 


■2156 


60.157 


.1823 


3-2 


•3125 


•394 




438 


1 .262 


1. 401 


69 


626 


73 • 4oo 


.2110 


.2224 


61.845 


.1874 


3-3 


• 3°3° 


.385 




428 


1.270 


i-4i4 


7i 


700 


75,686 


.2173 


.2294 


63.481 


.1924 


3-4 


.2941 


•376 




419 


1.277 


1.426 


73 


720 


77-936 


•2234 


.2362 


65.087 


.1972 


3-5 


•2857 


■ -3 6 7 




411 


1.285 


1.438 


75 


688 


80.131 


.2294 


.2428 


66.610 


.2019 


3-6 


.2778 


•359 




403 


1.292 


1.450 


77 


628 


82.307 


•2352 


.2494 


68.108 


.2064 


3-7 


.2703 


•35i 




395 


1.299 


1. 461 


79 


5i6 


84.411 


.2410 


•2557 


69.564 


.2108 


3-8 


.2632 


•343 




388 


1.306 


i-473 


81 


35° 


86.496 


.2465 


.2621 


70.982 


.2151 


3-9 


.2564 


•337 




381 


I-3I3 


1.484 


83 


158 


88.544 


.2520 


.2683 


72.364 


.2193 


4.0 


.2500 


■33° 




374 


i-3i9 


1-495 


84 


939 


90.510 


•2574 


•2 743 


73-7io 


.2234 


4.1 


.2439 


•323 




3 6 7 


1.326 


1.506 


86 


694 


92.472 


.2627 


.2802 


75-023 


.2274 


4.2 


.2381 


•3i7 




361 


1 -33 2 


1. 516 


88 


395 


94-434 


.2678 


.2862 


76.304 


.2312 


4-3 


.2326 


•311 




355 


i-339 


1.526 


90 


043 


96.346 


.2729 


.2919 


77-555 


•2350 


4-4 


.2273 


.306 




349 


1-345 


1-537 


9i 


691 


98.202 


•2779 


.2976 


78.776 


.2387 


4-5 


.2222 


.300 




344 


i-35i 


i-547 


93 


312 


100.012 


.2828 


•303 1 


79.972 


.2424 


4.6 


.2174 


•295 




338 


i-357 


i-557 


94 


882 


101.823 


•2875 


•3085 


81. 141 


•2459 


4-7 


.2128 


.290 




333 


*-3 6 3 


1.566 


96 


424 


103.616 


.2922 


.3140 


82.284 


•• 2494 


4.8 


.2083 


• 285 




328 


1.368 


1-576 


97 


966 


io5-37i 


.2969 


•3193 


83.404 


.2528 



129 



130 



COMPRESSED AIR 



Table I. — (Continued) 



I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


» 


4-9 
5-° 
5-i 


.2041 
.2000 
.1961 


.280 
.276 
.272 


•324 
•319 
•315 


i-374 

1.380 

i-385 


1.586 

i-595 
1.604 


99.481 
100.943 
102.405 


107. 109 
108. 811 
110.493 


•3015 
•3059 
•3103 


.3246 
•3297 
•3348 


84.50c 

85.574 
86.627 


.2561 

•2593 
•2625 


5-2 
5-3 
5-4 


.1923 
.1887 

.1852 


.267 
.263 

•259 


.310 
.306 
.302 


1 -391 
1.396 
1. 401 


1-613 
1.622 
1-631 


103.841 
105.260 
106.673 


112. 157 
113.830 
115.440 


•3147 
.3180 

•3 2 32 


•3398 

•3449 
•3498 


87.660 
88.673 
89.666 


.2657 
.2687 
.2717 


5-5 
5-6 
5-7 


.1818 
.1786 
•i754 


.256 
.252 
.248 


.298 

•294 
.291 


1.406 
1. 411 
1. 416 


1.640 
1.648 
1-657 


108.013 

io9-353 
110.683 


117. 010 
118.570 
120. 114 


•3273 
•33i4 

•3354 


•3546 
•3593 
•3640 


90.642 
9 1 . 600 
92.541 


.2747 
.2776 
.2805 


5-8 
5-9 
6.o 


.1722 
.1695 
.1667 


•245 
.242 
.238 


.287 
.284 
.280 


1. 42 1 
1.426 
i-43i 


1.665 
1.673 
1. 681 


112.003 
113-305 
114-581 


121.632 
123.150 
124.640 


•3394 
•3433 
•3472 


.3686 
•3732 
•3777 


93.466 
94-375 
95.27I 


•2833 
.2860 
.2887 


6.i 

6.2 

6-3 


.1639 
.1613 

•1587 


•235 
.232 
.229 


.277 

• 274 
.271 


1.436 
1.440 

i-445 


1.689 
1.697 

1-705 


115-831 
117.080 
118.303 


126. 113 

127.576 
129.030 


•35io 

•3548 
•3585 


.3822 
.3866 
.3910 


96.147 
97.012 
97.863 


.2914 
.2940 
.2966 


6. 4 

6-5 
6.6 


.1562 
•1538 
•1515 


.226 
.223 
.221 


.268 
• 265 
.262 


1.449 
1-454 
1.458 


*-7*3 
1. 72 1 
1.728 


H9-573 
120.723 
121.920 


130.466 
131.880 
133-30° 


.3622 
•3658 
•3694 


•3953 
•3997 
•4039 


98.700 

99-5 2 4 
100.336 


.2991 
.3016 
.3040 


6.7 
6.8 
6.9 


.1492 
.1471 
.1449 


.219 
.216 
.213 


•259 
.256 

•254 


1.464 
1.467 
1. 471 


1.736 
1.744 
1 -75i 


123.063 
124.205 
125.348 


134.710 
136.090 
I37-450 


•3729 
■3764 
•3799 


.4082 
.4124 
.4165 


101.134 
101.920 
102.700 


•3065 
.3088 
.3112 


7.0 

7-i 
7.2 


.1428 
.1408 
.1389 


.211 

.208 
.206 


.251 
.249 
.246 


1.476 
1.480 
1.484 


i-758 
1.766 

i-773 


126.492 
127.608 
128.708 


138.800 
140. 120 
141.430 


•3833 
.3867 
.3900 


.4206 
.4246 
.4286 


103.465 
104.219 
104.963 


•3135 
•3158 
.3181 


7-3 
7-4 
7-5 


•i37o 
•i35i 
-.1333 


.204 
.202 
.199 


• 244 
.241 

•239 


1.488 
1.492 
1.496 


1.780 
1.787 
1.794 


129.789 
130.878 
i3i-94i 


142.710 
!43-979 
145-239 


•3933 
.3966 

•399 8 


•4327 
■4363 
.4401 


105.696 
106.420 
107.133 


•3203 
•3225 
.3246 


7.6 

7-7 
7.8 


.1316 
.1299 
.1282 


.197 
•195 
•193 


•237 
•235 
•233 


1.500 
1.504 
1.508 


1. 801 
1.807 
1. 814 


I32-995 
134-043 
135-063 


146.489 
147.732 
148.976 


■ 4030 

.4062 

•4093 


•4439 

•4477 
•45 I 4 


107.837 
108.539 
109.219 


.3268 
•3289 
•33io 


7-9 
8.0 
8.1 


. 1266 
• 1250 
.1236 


.191 
.189 
.188 


.231 

.228 
.226 


1. 512 
1. 516 
i-5i9 


1. 821 
1.828 
1.834 


136.091 
137. no 
138. in 


150.217 
151.427 
152.633 


.4124 

•4i55 
.4185 


•4552 
•4589 
.4625 


109.896 
110.565 
in. 225 


•333° 
•335° 
•3370 


8.2 

8-3 
8.4 


. 1220 
.1205 
. 1 190 


.186 
.184 
.182 


.224 
.223 
.221 


I-523 
1-527 
i.-53i 


1. 841 

1.847 
1.854 


I39-093 
140.076 
141.060 


153-823 
155.010 
156.178 


•4215 
• 4245 
•4275 


.4661 
.4698 
•4733 


in. 875 
112.522 
113-158 


•3390 
.3410 

•3429 


8-5 
8.6 
8.7 


.1176 
.1163 
.1149 


.180 
.179 
.177 


.219 
.217 
.215 


i-534 
i-538 
1 -541 


1. 861 
1.867 
i-873 


142.017 
142.974 
I43.93I 


I57-348 
158.508 
159.658 


•4304 
•4333 
•4362 


.4768 
.4804 
•4838 


113.788 
114. 410 
115.023 


•3448 
•3465 
•3487 


8.8 
8. 9 
9.0 


.1136 
. 1124 
. 1111 


.176 
.174 
.172 


.214 
.212 
.210 


!-545 
1.548 

i-55 2 


1.879 
1.885 
1. 891 


144.862 
145.780 
146.700 


1 60 . 800 
161.927 
163.041 


•4390 
.4418 
• 4446 


•4873 
.4906 
.4941 


II5-633 
116.233 
116.827 


•3504 
•3522 
•3540 


9.1 

9.2 
9-3 


.1099 
.1087 
. 1072 


. 171 
.170 
.168 


.208 
.207 
•205 


i-555 
i-559 
1.562 


1.897 
1.903 
1.909 


147.627 

148.557 
149-554 


164.147 
165.236 
166.334 


•4474 
.4502 

•4532 


•4974 
•5007 
.5041 


ii7-4i5 
117.996 
118. 571 


•3558 
•3576 
•3593 


9.4 

9-5 
9.6 


. 1064 

.1058 
. 1042 


.167 
.165 
.164 


.204 
.202 
.201 


1-565 
1.569 

i-572 


i-9i5 
1. 921 
1.927 


150.312 
151. 188 
152.066 


167.431 

168.520 
169.589 


•4555 
.4582 
.4609 


•5074 
•5107 
•5139 


119-138 
119.702 
120.259 


.3610 
•3627 
•3644 


9-7 
9.8 
9.9 


■ 1031 

. 1020 

. IOIO 


. 162 
.161 
.160 


.199 
.198 
.196 


i-575i-933 
i-578i-939 
1. 5821. 944 


152.944 
153-794 
154-645 


170.650 
171.700 

I72.754 


•4635 
4661 
4686 


•5i7i 
•5213 
5235 


120.810 

121-355 
121.895 


.3661 
•3677 
•3693 


10. 


. IOOO 


•iS9 


• 195 


n. 5851. 950 


C55-495 


E 73. 789 


47 r2 


5266 


122.429 


.3710 



TABLES 



131 



Notes on Table II 

The purpose of this table is to determine the weight of air compressed by a 
machine of known cubic feet capacity. It is to be used in connection with Table 
I for determining power or work. 

The barometric readings and elevations are made out for a uniform tempera- 
ture of 6o°F. and are subject to slight errors but not enough to materially affect 
results. Table V gives more accurately the relation between elevation tem- 
perature and pressure. 



Table II.- 


-Weights of 


Free Air Under Various Conditions 


Approximate Baro- 
metric Reading. 
T=6o 


<u 
u 
3 

<u 
u 


'u 

<u 
J3 

p. 

E 
< 


Weight of One Cubic Foot at Given 
Temperature (Fahr.) 


a! 
> 

4) O 

w £ 
1 II 

1 6 

p. "^ 
< 


-20 


oo° 


20° 


40° 


6o° 


8o° 


IOO° 


I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


30-5 2 
3°-3 2 
30.12 


15.O 

14.9 

14-8 


.09211 
.09150 
.09089 


.08811 
•08753 

.08694 


. 08444 
.08388 
•08331 


.08108 
.08054 
. 08000 


.07796 
.07744 
.07693 


.07508 
.07458 
.07408 


.07240 
.07192 

.07144 


—600 
—400 
— 200 


29.91 
29.71 
29.50 


14.7 
14.6 

14-5 


.09027 
.08965 
.08903 


.08635 

.08576 
•08517 


•08275 
.08219 
.08163 


•07945 

.07895 
.07837 


.07640 
.07589 
•07536 


•073S8 

.07308 
.07258 


•07095 

•07047 

. 06999 


00 
200 
400 


29.30 
29.10 
28.90 


14.4 

14-3 
14.2 


.08842 
.08781 
.08719 


.08458 

.08400 

.08341 


.08107 
.0805c 
•o7994 


.07783 
.07729 
.07675 


.07484 

•07432 
.07380 


.07208 
.07158 
.07108 


.06950 

.06902 

.06854 


600 

800 

1000 


28.69 
28:49 
28.28 


14. 1 
14.0 

13-9 


.08659 
.08597 
•08535 


.08282 

.08224 

.08165 


.07938 
.07882 
.07825 


.07621 
•o75 6 7 
•07513 


.07329 
.07277 

.07225 


•07058 
.07008 
.06957 


. 06806 

•06758 

.06709 


1200 
1400 
1600 


28.08 
27.88 
27.67 


13.8 

13-7 
13.6 


.08474 
.08412 
.08351 


.08106 

.08048 

.07989 


.07769 

•07713 
■07656 


•07459 
.07405 

■07350 


•07173 
.07120 
.07068 


.06907 
.06857 
.06807 


.06661 
.06612 

.06564 


1800 
2000 
2100 


27.47 
27.27 
27.06 


i3-5 
13-4 
13-3 


.08289 
.08228 
.08167 


.07930 

.07871 

.07813 


.07600 
•07544 
•07487 


.07296 
.07242 
.07189 


.07016 
.06965 
.06913 


.06757 
.06707 
.06657 


.06516 

.06468 

.06420 


2300 
2500 

2700 


26.86 
26.66 
26.45 


13.2 

i3-i 

13.0 


.08106 
. 08044 
.07983 


•07754 
.07695 
.07637 


•07431 
•o7375 
.07319 


•o7i35 
.07080 
.07026 


.06861 

. 06809 
.06757 


.06607 

•o6557 
•06507 


.06371 
.06323 

.06274 


2900 
3100 

33°° 


26.25 
26.05 

25.84 


12.9 
12.8 
12.7 


.07921 

.07860 
.07798 


.07578 
■07518 

.07460 


.07262 
.07206 
.07150 


.06972 
.06918 
.06862 


•06705 
.06652 
. 06600 


•06457 

.06407 
•06357 


.06226 

.06178 

.06130 


35°° 
3700 
4000 


25.64 
25-44 
25-23 


12.6 
12.5 

12.4 


•07737 
.07676 
.07615 


.07401 

■07343 

.07284 


.07094 

.07038 
.06981 


.06810 
.06756 
.06702 


.06549 
.06497 
.06445 


.06307 

•06257 
.06207 


.06082 

•06033 
•05985 


4200 
4400 
4600 



132 



COMPRESSED AIR 









Table II- 


-{Continued) 








I 2 


3 


4 


5 


6 


7 


8 


9 


10 


25-°3 
24.83 
24.62 


12.3 
12.2 
12. 1 


•07553 
.07492 
.07430 


•07225 
.07166 
.07108 


.06925 
.06868 
.06812 


. 06648 
.06594 
.06540 


• 06393 
.06341 
.06289 


.06157 
.06107 
.06057 


•05937 
.05889 
• 05840 


4800 
5000 
5200 


24.42 
24.22 
24.01 


12.0 
11. 9 
11. 8 


.07369 

.07307 
.07246 


.07049 
. 06990 
.06932 


.06756 
. 06699 
. 06643 


.06486 
.06432 
.06378 


.06237 
.06185 
.06133 


.06007 

•05957 
.05907 


.05792 

•05744 
.05696 


5400 
5600 
5800 


23.81 
23.60 
23.40 


11. 7 
11. 6 

ii-S 


.07184 
.07123 
.07061 


•06873 
.06812 
• o6 755 


.06587 
.06530 
.06474 


.06324 
.06270 
.06216 


.06081 
.06029 
•05977 


•05857 
.05807 

•05757 


•05647 
■05599 
•o555i 


6100 
6300 
6500 


23.20 
22.99 
22.79 


11. 4 

"•3 

11. 2 


.07000 
.06938 
.06877 


. 06693 
.06638 
•06579 


.06418 
.06362 
.06305 


.06161 
.06108 
.06054 


•05925 
•05873 
.05821 


.05707 
.05656 
.05606 


.05502 

•05454 
-05406 


6800 
7100 
7300 


22.59 
22.38 
22.18 


11. 1 
11. 
10.9 


.06816 
.06754 
.06692 


.06520 
.06462 
.06403 


.06249 
•06193 
.06136 


. 06000 

•o5945 
.05891 


■05769 
•05717 
.05665 


•05556 
•05506 
•05456 


■0535 8 
•05310 
.05261 


7600 
7900 
8100 


21.98 

21.77 
21-57 


10.8 
10.7 
10.6 


.06632 
•06571 
.06510 


•06344 
.06285 
.06226 


. 06080 
.06024 
.05968 


•05837 
•05783 
.05729 


•05613 

•05561 
•05509 


.05406 
•05356 
.05306 


•05213 
•05164 
.05116 


8400 
8600 
8900 


21.37 
21.16 
20.96 


10.5 
10.4 

10.3 


. 06448 
.06386 
.06325 


.06168 

.06109 
.06050 


.05911 

•05855 
•o5799 


•05675 
.05621 

•05567 


•o5457 
•05405 
•o5353 


•05256 
.05206 
•05156 


.05068 
.05020 
.04972 


9100 
9400 
9600 


20.76 
20.55 
20.35 


10.2 
10. 1 
10. 


.06263 
.06202 
.06141 


.05991 

•05933 
.05874 


•o5743 
.05686 
.05630 


•05513 
•05459 
•05405 


•05301 
•05249 
•05198 


.05106 
.05056 
.05006 


.04923 

.04875 
.04827 


9900 

IOIOO 

10400 


20. 15 
19-94 
19.74 


9.9 
9.8 
9-7 


.06079 
.06017 
■05956 


.05816 

•o5757 
.05698 


•05572 
•05517 
.05461 


■05351 
.05297 

•05243 


.05146 
.05094 
.05041 


•04956 
. 04906 
.04856 


.04779 

■04730 
.04682 


10700 
I IOOO 
1 1200 


19-53 

^ 19-33 

I9-I3 


9.6 
9-5 
9-4 


.05894 

•05833 
•05772 


•05639 

.05580 
•05522 


.05404 
•05348 
.05292 


.05188 

•05134 
.05081 


. 04990 

•04937 
.04886 


.04806 
•04756 
.04706 


• 04633 
•04585 
•0453 8 


1 1 500 

1 1 800 

I2IOO 


18.93 
18.72 
18.52 


9-3 
9.2 
9.1 


.05711 
■05649 
•05587 


•05463 
•05404 
•05345 


.06236 

•05179 
•05123 


•05027 
.04972 
.04918 


• 04834 
.04782 

•04730 


•04655 
.04605 

•04555 


. 04489 
. 04440 
.04392 


I24OO 
I270O 
13OOO 


1S.31 


9.0 


■05526 


.05286 


.05067 


.04864 


.04678 


•04505 


• 04344 


I34OO 



Note on Table III 
The table is designed to compute readily weights of compressed air by formula 



12, Art. 8, viz., w 



becomes w 



53-17 * 
144 X p 
53-^7 X t 



If p is given in pounds per square inch the formula 



Table III. — Weights of Compressed Air 
Pounds per Cubic Foot 

P 
The Ratio - is for absolute pressure in pounds per square inch and absolute 

temperature Fahrenheit. (See Note at foot of previous page.) 



i 
t 


w 


P 
t 


w 


t 

t 


w 


t 


W 


.000 

.005 

.010 


0000 

0135 

0271 


•255 
.260 
.265 


.6906 
.7041 
.7177 


5io 
515 

520 


1-3813 

1-3947 
1 . 4083 


765 
77o 

775 


2.0718 
2.0853 
2.0988 


.015 
.020 
.025 


0406 

0542 

0677 


.270 

•275 
.280 


•7312 
•7447 
•7583 


525 
53° 
535 


1 .4219 

1-4355 
1 . 4490 


780 

785 

79o 


2-1125 
2. 1260 
2-1395 


.030 

•035 
.040 


0813 

0948 

1083 


.285 
.290 
•295 


.7719 

•7852 
.7989 


54o 
545 
55o 


1.4625 
1.4760 
1.4895 


795 
800 
805 


2-153° 
2. 1665 
2.1798 


■045 
.050 

•055 


1218 

1354 
1489 


.300 

•3°5 
.310 


.8125 
.8260 
•8395 


555 
560 

565 


1 ■ 5°3° 
1. 5166 
i-53i2 


810 

815 
820 


2.1950 
2.2071 
2 . 2207 


.060 
.065 
.070 


1625 

1760 

1896 


•315 
.320 

•325 


•8531 
.8666 
.8801 


57° 
575 
580 


1-5437 
I-557 2 
i-57°7 


825 

830 

835 


2-2343 

2 . 2480 
2.2615 


•075 
.080 
.085 


2031 

2166 
2302 


■33° 
•335 
•34o 


•8937 
.9072 
.9208 


585 
59° 
595 


1 ■ 5843 
1.5980 
1-6115 


840 

845 
850 


2.2750 
2.2885 
2 .3020 


.090 

'•095 
. 100 


2437 
2573 

2708 


•345 
•35o 

•355 


•9343 
.9478 
.9613 


600 
605 
610 


1.6250 
1.6385 
1.6520 


855 
860 
865 


2-3155 
2.3290 

2-3425 


.105 

. no 

■115 


2843 
2979 

3"4 


.360 
■3°5 

•37° 


•9749 

.9S84 

1.0020 


615 
620 
625 


1 . 6654 
1.6792 
1.6927 


870 

875 
880 


2-356i 
2.3698 

2-3 8 33 


. 120 

■ 125 
.130 


3250 
3385 
352° 


•375 
.380 

•385 


i-oi55 
1 .0290 
1.0425 


630 

635 
640 


1 . 7062 
1. 7198 
1-7333 


885 
890 
895 


2.3970 
2.4105 
2.4240 


• 135 

.140 

• 145 


3656 
3792 
3927 


•39° 
■395 
.400 


1. 0561 
1.0697 
1-0833 


645 
650 

655 


1 . 7468 
1 . 7603 
1-7739 


900 

905 
910 


2-4375 
2.4510 

2 ■ 4645 


.150 
• 155 

. 160 


4062 
4197 
4333 


•405 
.410 

•415 


1 .0968 
1. 1 103 
r. 1240 


660 
665 
670 


I-7875 
1 .8010 

1-8145 


915 
920 

925 


2.4780 
2.4917 
2.505 2 


.165 

.170 

• 175 


4468 
4603 
4739 


.420 

•425 
•43° 


1 • 1375 
1. 1510 

1-1645 


675 
680 
685 


1.8280 
1-8415 
1-8550 


930 
935 
940 


2.5187 
2-5323 
2-5459 


.180 

.185 

.190 


4875 
5010 

5145 


•435 
.440 

•445 


1 . 1 780 
1 . 1917 
1.2052 


690 

695 
700 


1 . 8680 
1.8822 
1.8959 


945 
95o 
955 


2 5594 
2-573° 
2.5865 


• 195 

.200 

.205 


5281 
54i6 
555i 


•45° 
•455 
.460 


1 .2177 
1.2323 

1-2457 


705 
710 

7i5 


1 . 9094 
1.9229 
1-9365 


960 

965 
970 


2 . 6000 
2-6135 

2 . 6270 


. 210 

• 215 

.220 


5687 
5822 
•5958 


•465 
.470 

•475 


1-2594 
1.2730 
1.2865 


720 

7- 7 5 
73° 


1.9500 

;-9G35 

1.9770 


975 
980 

985 


2 ■ 6405 
2.6541 
2 6670 


.225 
.230 
• 235 


.6094 
6229 
.6364 


.480 

.485 
.490 


1 . 3000 

i-3i35 
1.3270 


735 
740 

745 


1 9905 
2 . 0042 
2.0177 1 


990 

995 

000 


2.6813 

2 . 6949 
2 7084 


.240 

• 245 

.250 


• 6499 
6635 
6771 


•495 
.500 

■ 505 


1. 3416 
1-3542 
1-3677 


75o 
755 

760 


2.0312 

2.0448 
2.0582 







133 



Table Ilia— 


Giving The Values of "K" and "H" Corresponding to 


Each 


2 «. 

3 M 'S 

u so <u 
BO'S 


<u 
3 

> 


m 

O 

to 

CU 

3 

> 


3 CO CO 
4-> CU <-< 

h t" £ 

BO'S 


a 



CO 

cu 

3 
> 


O 
3 

> 


CO 

^ +3 
3 to co 

p. i) b 



to 

co 

> 


&3 
O 

<u 

3 

> 


CD 

<-■ *3 

3 to'S 

2 "c 
2 &>£ 

SQ-S 

cu cL 


O 

a) 

"(4 
> 




S 

> 




u f! 
3 co'S 

■P u S 

cu ho <u 

SQ-S 

CU Cl, 


M 



CO 

cu 

3 

*CCi 

> 


&3 


3 
"3 

> 


-30 


.6082 


.0099 


17 


.6132 


.0941 


64 


.6188 


.5962 


III 


.6251 


2.654 


158 


■6323 


9.177 


-29 


.6083 


.0105 


18 


.6133 


.0983 


65 


.6189 


.6175 


112 


.6253 


2.731 


159 


.6325 


9.400 


-28 


.6084 


.0111 


19 


.6134 


.1028 


66 


.6190 


■ 6393 


113 


• 6255 


2. 811 


160 


.6326 


9.628 


-27 


.6085 


.0117 


20 


• 6135 


.1074 


67 


.6192 


.6617 


114 


.6256 


2.892 


161 


.6328 


9.860 


-26 


.6086 


.0123 


21 


.6136 


.1122 


68 


.6193 


.6848 


115 


.6257 


2.976 


162 


.6330 


10.10 


-25 


.6087 


.0130 


22 


■ 6137 


.1172 


69 


.6194 


.7086 


Il6 


.6258 


3.061 


163 


.6331 


10.34 


-24 


.6088 


.0137 


23 


■ 6139 


.1224 


70 


.6196 


■ 7332 


117 


.626O 


3.149 


164 


.6333 


10.59 


-23 


.6089 


.0144 


24 


.6140 


.1279 


7i 


.6197 


.7585 


118 


.6261 


3.239 


165 


.6335 


10.84 


— 22 


.6090 


.0152 


25 


.6141 


.1336 


72 


.6198 


.7846 


119 


.6263 


3-331 


166 


.6336 


n. 10 


— 21 


.6091 


.0160 


26 


.6142 


.1396 


73 


.6199 


.8114 


120 


.6264 


3.42s 


167 


.6338 


n .36 


— 20 


.6092 


.0168 


27 


.6143 


.1458 


74 


.6201 


.8391 


121 


.6266 


3.522 


168 


.6340 


11.63 


-19 


.6093 


.0177 


28 


.6144 


.1523 


75 


.6202 


.8676 


122 


.6267 


3.621 


169 


■ 6341 


11 .90 


-18 


.6094 


.0186 


29 


.6146 


• 1590 


76 


.6203 


.8969 


123 


.6269 


3.722 


170 


■ 6343 


12.18 


-1.7 


.6095 


.0196 


30 


.6147 


.1661 


77 


.6205 


.9271 


124 


.6270 


3.826 


171 


• 6345 


12.46 


-16 


.6096 


.0206 


31 


.6148 


■1734 


78 


.6206 


.9585 


125 


.6272 


3.933 


172 


■ 6346 


12. 75^ 


-15 


.6097 


.0216 


32 


.6149 


.l8ll 


79 


.6207 


.9906 


126 


.6273 


4.042 


173 


.6349 


13.04 


-14 


.6098 


.0227 


33 


.6150 


.1884 


80 


.6209 


I .024 


127 


.6275 


4-153 


174 


.6350 


13.34 


-13 


.6099 


.0238 


34 


.6151 


.i960 


81 


.6210 


1. 057 


128 


.6276 


4.267 


17s 


.6352 


13.65 


— 12 


.6100 


.0250 


35 


.6153 


.2039 


82 


.6211 


1.092 


129 


.6278 


4.384 


176 


.6353 


13.96 


— II 


.6101 


.0262 


36 


.6154 


.2120 


83 


.6213 


1. 128 


130 


.6279 


4-503 


177 


■ 6355 


14.28 


— 10 


.6102 


.0275 


37 


.6155 


.2205 


84 


.6214 


1. 165 


131 


.6281 


4-625 


178 


• 6357 


14.60 


- 9 


.6103 


.0289 


38 


.6156 


.2292 


85 


.6215 


1.203 


132 


.6282 


4-750 


179 


.6359 


14.92 


- 8 


.6104 


.0303 


39 


■ 6157 


.2382 


86 


.6217 


1.242 


133 


.6284 


4.877 


180 


.6360 


15.27 


- 7 


.6105 


.0317 


40 


.6158 


.2476 


87 


.6218 


1.282 


134 


.6285 


5.008 


181 


.6362 


15.62 


- 6 


.6107 


.0332 


41 


.6160 


.2572 


88 


.6219 


1.324 


135 


.6287 


5.142 


182 


■ 6364 


15.97 


- 5 


.6108 


.0348 


42 


.6161 


.2673 


89 


.6221 


1.366 


136 


.6288 


5.280 


183 


.6365 


16.32 


- 4 


.6109 


.0365 


43 


.6162 


.2776 


90 


.6222 


1 .410 


137 


.6290 


5. 420 


184 


.6367 


16.68 


- 3 


.6110 


.0382 


44 


.6163 


.2883 


9i 


.6223 


1.455 


138 


.6291 


5.563 


185 


.6369 


17.05 


— 2 


.6111 


.0400 


45 


.6164 


.2994 


92 


.6225 


1. 501 


139 


.6293 


5.709 


186 


• 6371 


17-43 


— I 


.6112 


.0419 


46 


.6166 


.3109 


93 


.6226 


I.S48 


140 


.6294 


5-859 


187 


.6373 


17.81 





.6113 


0439 


47 


.6167 


.3227 


94 


.6227 


1.597 


141 


.6296 


6. on 


188 


.6374 


18.20 


+ I 


.6114 


.0459 


48 


.6168 


• 3350 


95 


.6229 


1.647 


142 


.6298 


6.167 


189 


.6376 


18.59 


2 


.6115 


.0481 


49 


.6169 


■ 3477 


96 


.6230 


1.698 


143 


.6299 


6.327 


190 


• 6377 


19.00 


3 


.6116 


0503 


50 


.6170 


.3608 


97 


.6232 


r:75i 


144 


.63OI 


6.490 


191 


.6380 


19.41 


4 


.6117 


.0526 


5i 


.6172 


■ 3743 


98 


■6233 


1.805 


145 


.6302 


6.656 


192 


.6381 


19.83 


5 


.6118 


.0551 


52 


.6173 


.3883 


99 


.6234 


1. 861 


I46 


.6304 


6.827 


193 


.6383 


20.25 


6 


.6120 


.0576 


53 


.6174 


.4027 


100 


.6236 


1. 918 


147 


■6305 


7 .001 


194 


.6385 


20.69 


7 


.6121 


.0603 


54 


■ 6175 


.4176 


101 


.6237 


1.976 


I48 


.6307 


7.178 


195 


.6387 


21.13 


8 


.6122 


.0630 


55 


.6177 


■ 4331 


102 


.6238 


2.036 


149 


.6309 


7-359 


196 


.6389 


21.58 


9 


.6123 


0659 


56 


.6178 


.4490 


103 


.6240 


2 .098 


150 


.63IO 


7-545 


197 


.6391 


22 .04 


10 


.6124 


.0690 


57 


.6179 


.4655 


104 


.6241 


2 .161 


151 


.6312 


7-736 


198 


.6393 


22.50 


ii 


.6125 


.0722 


58 


.6180 


.4824 


105 


.6243 


2.226 


152 


■6313 


7.929 


199 


■ 6394 


22.97 


12 


.6126 


0754 


59 


.6182 


• 4999 


106 


.6244 


2 .294 


153 


.6315 


8.127 


200 


.6396 


23.46 


13 


.6127 


0789 


60 


.6183 


.5180 


107 


.6246 


2.362 


154 


.6317 


8.328 


201 


.6397 


23-94 


14 


.6128 


.0824 


61 


.6184 


■ 5367 


108 


.6247 


2.432 


155 


.6318 


8.534 


202 


.6400 


24.44 


15 


.6130 


.0862 


62 


.6185 


■ 5559 


109 


.6248 


2.504 


156 


.6320 


8.744 


203 


.6402 


24-95 


16 


.6131 


.0900 


63 


.6187 


■ 5758 


no 


.6250 


2.578 


157 


.6322 


8.958 


204 


.6404 


25-47 



134 



Fahrenheit Degree of 


Temperature From 30 Below to 434 Above Zero 


■^ <D r* 
c! (D-S 

h »-• s 

« a? 
ft" S 


O 

0> 


&3 


en 

<o 

51 
*c3 


Oj CD'S 

u bo, 1 " 

BO'S 


O 

*c3 


51 

*3 


a w 3 
6Q^ 


*4-t 
O 

51 
13 


&3 

5) 


u +; 

5* tn a) 

gQ-S 


*4 

O 

<u 
5) 

"(3 




51 


•* l- 1 K 

ft»5 

gQ-S 




CO 

<u 

51 

*c4 


"o 
13 




> 


> 


CD CU 


> 


> 




> 


> 




> 


> 




> 


> 


205 


1 
.640525.99 


251 


.6499,61 .89 


297 


.6607 


130.8 


343 


.6736 


250.9 


389 


.6890 


444-4 


206 


.640726.53 


252 


.6501,62.97 


298 


.6610 


132.8 


344 


.6739 


254.2 


390 


■ 6893 


449.6 


207 


.640927.07 


253 


.650364.08 


299 


.6612 


134-8 


345 


.6741 


257.6 


391 


.6897 


454-9 


208 


.6411 27 .62 


254 


.650565.21 


300 


.6615 


136.8 


346 


.6745 


261 .0 


392 


.6901 


460.2 


209 


.641328.18 


255 


.6508 


66.34 


301 


.6617 


138.9 


347 


.6749 


264.5 


393 


.6905 


465.6 


210 


.6415 


28.75 


256 


.6510 


67.49 


302 


.6620 


141 .0 


348 


■ 6751 


268.0 


394 


.6908 


47O.9 


211 


.6417 


29.33 


257 


.651268.66 


303 


.6623 


143. 1 


349 


■ 6754 


271.5 


395 


.69H 


476.4 


212 


.6419 


29.92 


258 


.651469.85 


304 


.6625 


145.3 


350 


.6757 


275.0 


396 


.6915 


48I.9 


213 


.6421 


30.53 


259 


.6516 


71. OS 


305 


.6628 


147.4 


351 


.6760 


278.6 


397. 


.6919 


487.4 


214 


.6423 


31.14 


260 


.6518 


72 .26 


306 


.6631 


149.6 


352 


.6763 


282.2 


398 


.6923 


493-0 


215 


.6424 


31.76 


261 


.6521 


73.50 


307 


.6633 


151. 8 


353 


.6767 


285.9 


399 


.6927 


498.7 


216 


.6426 


32.38 


262 


.6523 


74-75 


308 


.6636 


I54-I 


354 


.6770 


289.6 


400 


.6931 


504.4 


217 


.6428 


33-02 


263 


.6525 


76.02 


309 


.6639 


156.3 


355 


.6773 


293-3 


401 


.6935 


510. 1 


218 


.6430 


33.67 


264 


.6528 


77-30 


310 


.6641 


158.7 


356 


.6776 


297.1 


402. 


.6939 


515.9 


219 


.6432 


34-33 


265 


.6530 


78.61 


311 


.6644 


161. 


357 


.6780 


300.9 


403 


•6943 


521. 7 


220 


.6434 


35-01 


266 


.6532 


79-93 


312 


.6647 


i63-3 


358 


.6783 


304.8 


404 


.6947 


527.6 


221 


.6436135-69 


267 


.6534 


81.27 


3i3 


.6650 


165.7 


359 


.6786 


308.7 


40s 


.6951 


533-5 


222 


.643836.38 


268 


.6537 


82.62 


314 


.6652 


168. 1 


360 


.6789 


312.6 


406 


.6955 


539-5 


223 


.6440,37.08 


269 


-6539 


84.00 


3i5 


.6655 


170.5 


36l 


.6792 


316.6 


407 


.6958 


545.6 


224 


.6442 


37.80 


270 


.6541 


85.39 


316 


.6658 


173-0 


362 


.6795 


320.6 


408 


.6962 


551.6 


225 


.6444 


38.53 


271 


• 6543 


86.83 


3i7 


.6661 


175-5 


363 


■ 6799 


324.6 


409 


.6966 


557.8 


226 


.6446 ! 39.27 


272 


.6546 


88.26 


318 


.6663 


178.0 


364 


.6803 


328.7 


410 


.6970 


564.0 


227 


.644840.02 


273 


.654889.71 


319 


.6666 


180.6 


365 


.6806 


332.8 


411 


.6975 


570.2 


228 


.645140.78 


274 


.6551 


91.18 


320 


.6669 


183. 1 


366 


.6809 


337-0 


412 


.6979 


576.5 


229 


.6453 41.56 


275 


.6553 


92.67 


321 


.6671 


185.7 


367 


.6813 


341-2 


413 


.6983 


582.8 


230 


.645542.34 


276 


.6555 


94-18 


322 


.6674 


188.3 


368 


.6816 


354-4 


414 


.6987 


589.3 


231 


.6457 43.14 


277 


■655895.71 


323 


.6677 


191. 


369 


.6820 


349-7 


415 


.6991 


595-7 


232 


.6458 


43-95 


278 


.656097 .26 


324 


.6680 


193.7 


370 


.6822 


354-0 


416 


.6995 


602 .2 


233 


.6460 


44-77 


279 


.656398.83 


325 


.6683 


196.5 


371 


.6825 


358.4 


417 


.6999 


608.8 


234 


.6463 


4S-6i 


280 


.6565 


100.4 


326 


.6686 


199-2 


372 


.682g 


362.8 


418 


■7003 


615.4 


235 


.6465 


46.46 


281 


.6568 


102.0 


327 


.6689 


202.0 


373 


.6832 


367.3 


419 


.7007 


622.1 


236 


.6467J47.32 


282 


■ 6570 


103.7 


328 


.6691 


204.8 


374 


.6836 


371.8 


420 


.7012 


628.8 


237 


.6469148.19 


283 


■ 6572 


105-3 


329 


.6694 


207.7 


375 


.6839 


376.3 


421 


.7016 


635.6 


238 


.647149.08 


284 


• 6575 


107.0 


330 


.6697 


210.5 


376 


.6843 


380.9 


422 


.7021 


642.5 


239 


.6473 


49.98 


285 


■ 6577 


108.7 


33i 


.6700 


213-5 


•377 


.6847 


385.5 


423 


.7025 


649.4 


240 


.6475 


50.89 


286 


.6580 


no. 4 


332 


.6703 


216.4 


378 


.685c 


390.2 


424 


.7029 


656.3 


241 


.6477 


51.83 


287 


.6582 


112. 1 


333 


.6707 


219.4 


379 


.6853 


394-9 


425 


.7033 


663.3 


242 


.6479 


52.77 


288 


.6584 


II3-9 


334 


.6709 


222.4 


380 


.6857 


399-6 


426 


.7037 


670.4 


243 


.6481 


53-72 


289 


.6587 


II5- 8 


335 


.6712 


225.4 


381 


.6861 


404-3 


427 


.7042 


677.5 


244 


.6484 


54-69 


290 


.6590 


II7-5 


336 


.6715 


228.5 


382 


.6865 


409.3 


428 


.704C 


684.7 


245 


.6486 


55-68 


291 


.6592 


119- 3 


337 


.6717 


231.6 


383 


.6868 


414.2 


429 


.7051 


691.9 


246 


.6488 


56.67 


292 


.6594 


121 .2 


338 


.6721 


234-7 


384 


.6871 


419. 1 


430 


.7055 


699-2 


247 


.6490 


57.69 


293 


■ 6597 


123. 1 


339 


.6724 


237-9 


385 


.6875 


424.1 


431 


• 705C 


706.5 


248 


.6492 


58.71 


294 


.6600 


125.0 


340 


.6727 


241 .1 


386 


.687? 


429 .1 


432 


.7064 


713-9 


249 


.6494 


59.76 


295 


.6602 


126.9 


341 


.6730 


244-3 


387 


.6882 


434-2 


433 


.7068 


721.4 


250 


.649660.81 


296 


.6604 


128.8 


342 


• 6733 


247-6 


388 


.688C 


439-3 


434 


.7073728.9 



135 



136 



COMPRESSED AIR 



Table IV.* — Special Table Relating to Stage Compression From 

Free Air at 14.7 Pounds Pressure and 62 Temperature 

Compression adiabatic but cooled between stages 





H 

3 

<u 

In 


c 


P. 
| 


Is 

" I* 

a) § . 


Single Stage 


Two Stage 





3 


1 «M ■» 

e ° -3 

-u a 
fe s 
d ^ 


a 


a 

V o 

ft CTj 


•Si 

3 
+-> 


1 *w 0) 

a ° 3 

4J b 

•2 3 


ft 
t) 

60 
n! 
O 


3 

.9 


•5 i, ° 3 
.5? En -2 

4) 


1 8 


a -a 


* a.* 

w S 11 

0) 0) «> 


Is 

• 2 a 

+> .H 


S W 4, 

H •« 1 

B W g 


£ B 1- 

0) 0) <u 

1 ft£ 






£ 


a 


E 


M 


« 


E 


a 


Pg 


r 


w 


M.E.P. 


Tx 


H.P. 


Vr 


r 2 


H.P. 


5 


i-34 


. 1020 


4-5° 


108 


.0197 








10 


1.68 


.1279 


8.30 


144 


.0362 








15 


2.02 


•1537 


n. 51 


177 


.0045 








20 


2.36 


.1796 


14.40 


207 


.0628 








2 5 


2.70 


•2055 


17.00 


235 


.0742 








3° 


3-°4 


•2313 


19.40 


259 


.0845 








35 


3-38 


•2572 


21.65 


280 


.0944 








40 


3-72 


.2831 


23.60 


303 


.1030 








45 


4.06 


.3090 


2 5-5° 


321 


.1112 








5° 


4.40 


•3348 


27.50 


341 


■"95 


2. 10 


180 


.1063 


55 


4-74 


.3607 


29. 10 


358 


.1268 


2. 17 


189 


.1123 


60 


5.08 


.3866 


30.75 


373 


•1339 


2.25 


196 


.1184 


65 


5-42 


.4124 


3 2 -3° 


392 


.1408 


2-33 


200 


• 1235 


. 70 


5-76 


.4383 


33- 80 


405 


.1472 


2.40 


207 


.1286 


75 


6.10 


.4642 


35-i8 


420 


•1532 


2-47 


214 


.1329 


80 


6.44 


.4901 


36.55 


434 


.1590 


2-54 


222 


.1372 


85 


6.78 


•5i59 


37-9o 


447 


.1650 


2.60 


227 


.1410 


90 


7. 12 


.54i8 


39.10 


461 


■1705 


2.67 


233 


. 1462 


95 


7.46 


.5676 


40.35 


473 


•1758 


2-73 


238 


.1500 


100 


7.80 


•5935 


41.65 


485 


.1812 


2.79 


242 


.1542 


i°5 


8.14 


.6194 


42.30 


497 


.1841 


2.85 


246 


•1578 


no 


8.48 


•6453 


43 • 75 


508 


.1908 


2.90 


251 


.1615 


115 


8.82 


.6712 


45.16 


5i9 


.1965 


2.99 


256 


.1648 


120 


9.16 


.6971 


46.00 


530 


.2008 


3-02 


259 


.1681 


125 


9-5° 


.7230 


47-o5 


54o 


• 2045 


3.08 


262 


. 1710 


130 


9.84 


.7488 


47.80 


55o 


.2085 


3-i4 


266 


.1740 


135 


10.18 


•7747 


48.85 


560 


•2135 


3-i9 


269 


• 1775 


140 


10.52 


.8005 


49.90 


5 6 9 


.2176 


3-24 


272 


.1810 


145 


10.86 


.8264 


51.00 


578 


.2220 


3- 2 9 


276 


• 1837 


i5° 


11.20 


.8522 


5i-7o 


587 


•2255 


3-35 


280 


.1865 



* The table is limited to the special initial condition of air specified in the 
caption. The assumption of 14.7 as atmospheric pressure makes the weights 
and work a little in excess of average conditions. However, it is a valuable and 
very instructive table. 



TABLES 



137 







Table IV.- 


— (Cont 


inued) 












s 


.« fa 

,Q O 


T 


wo Stage 


Three Stage 


a 




■BJ: 


1 <W <U 


a 



-Si 


1 MH V 

S ° 3 






3 N 




V J& 


Owe 




u "J 


O 4J C 




l-i 




1- so 
ft nj 


K fa 
3 


<-> fe a 
3 s 


<8 v 

u bo 
ft cS 




O fa ~ 
O ^ 


3 

o 
u 

fa 


s 







"3 

4J +3 




°fa* 




fe C h 

O < 

fa 1/1 
10 <u 


O 

O W 


ft 5 


S3 <o » 
O << 

fa HI 


60 




M 
'S fa 


3.fl 


■3 rt 5 


(t> <y flj 

1 &£ 




0) '" 


"3 "S c 


0) 4> <U 


® 


fa 


fa - 


£ 


K 


rt 


S 


w 


Pg 


r 


Ztf 


(r? 


r 2 


H.P. 


(r) 1 * 


r 3 


H.P. 


IOO 


7.8 


•5936 


2.79 


242 


.1542 


1.98 


176 


.1450 


150 


11. 2 


.8522 


3 


35 


280 


.1865 


2 


24 


200 


•1752 


200 


14.6 


I.IIIO 


3 


82 


3 os 


.2110 


2 


44 


215 


.1965 


250 


18.0 


1-3697 


4 


24 


332 


•2315 


2 


62 


226 


.2140 


300 


21.4 


1.6285 


4 


63 


353 


.2490 


2 


78 


241 


.2295 


350 


24.8 


1.8872 


4 


98 


37° 


.2640 


2 


92 


251 


.2418 


400 


28.2 


2.1459 


5 


31 


386 


.2770 


3 


04 


259 


•2535 


45° 


31.6 


2 . 4048 


5 


61 


399 


.2895 


3 


16 


267 


.2630 


500 


35-° 


2 . 6634 


5 


91 


412 


.2915 


3 


27 


275 


.2730' 


55° 


38.4 


2.9221 








3 


37 


281 


.2830 


600 


41.8 


3.1810 








3 


47 


287 


.2910 


650 


45-2 


3-4395 








3 


56 


292 


.2960 


700 


48.6 


3 • 6982 








3 


64 


297 


•3°25 


75o 


52.0 


3-957° 








3 


73 


302 


.3090 


800 


55-4 


4-2155 








-3 


80 


3°7 


•315° 


850 


58.8 


4-4745 








3 


83 


312 


.3210 


900 


62 .2 


4-733° 








3 


96 


316 


.3260 


95° 


65.6 


4.9920 








4 


°3 


320 


•3315 


1000 


69.0 


5-251° 








4 


10 


324 


•336o 


1050 


72.4 


5 ■ 5°95 








4 


17 


328 


.3400 


I IOO 


75-8 


5 - 7684 








4 


23 


33i 


•3445 


1150 


79.2 


6.0270 








4 


29 


334 


•349° 


1200 


82.6 


6.2855 








4 


36 


337 


•3525 


1250 


86.0 


6-5445 








4 


4i 


34i 


•357° 


1300 


89.4 


6 . 8030 








4 


47 


344 


•3615 


!35° 


92.8 


7.0620 








4 


52 


347 


.3660 


1400 


96.2 


7.3210 








4 


58 


35° 


•3685 


145° 


99.6 


7-5795 








4 


64 


353 


.3710 


1500 


103.0 


7.8382 








4- 


70 


356 


•3740 


i55o 


106.4 


8.0965 








4 


75 


359 


.3780 


1600 


109.8 


8-355° 








4- 


79 


361 


.3820 


1650 


113. 2 


8.6140 








4 


83 


363 


•385° 


1700 


116. 6 


8.8730 








4 


87 


365 


.3880 


i75o 


120.0 


9.1320 








4 


93 


367 


•39i5 


1800 


123.4 


9 • 39°° 








4 


97 


369 


•394o 


1850 


126.8 


9 • 6485 








5 


02 


37i 


•3965 



138 



COMPRESSED AIR 



Table V. — Varying Pressures with Elevations 
Solution of formula 20, Art. 17, viz. logi pa = 1. 16866 — 



h 



122.4 1 





Pressure 


n Pounds per Square Inch 


Elevation in Feet 










Temp. 50 F. 


Temp 35 F. 


Temp. 2o°F. 





14.70 


14.70 


14.70 


1000 


14.17 


14-15 


14 


14 


2000 


13.66 


I3 6 3 


13 


99 


3000 


13.16 


13.12 


13 


07 


4000 


12. 69 


12.63 


12 


57 


5000 


12.23 


12. 16 


12 


09 


5280 


12. 10 


12.03 


II 


96 


6000 


II.78 


II . 71 


II 


63 


7000 


II.36 


II.27 


II 


18 


8000 


IO.95 


IO.85 


IO 


75 


9000 


io-SS 


IO.45 


IO 


33 


1 0000 


10. 17 


IO.06 


9 


94 


12500 


9.28 


915 


9 


02 


15000 


8.46 


8.32 


8 


18 



Table VI.* — -Highest Limit to Efficiency When Compressed Air is Used 

Without Expansion, Assuming Atmospheric Pressure = 14.5 

Pounds per Square Inch 



r 


h 


E 


r 


h 


E 


r 


h 


E 


1.2 


6.66 


91.4 


5- 2 


140.0 


49 -o 


9.2 


273-3 


40. 2 


1.4 


*3-3 


84.9 


5-4 


146.6 


48.3 


9.4 


280.0 


39 


9 


1.6 


20.0 


79.8 


5-6 


153-3 


47-7 


9.6 


286.6 


39 


6 


1.8 


26.6 


75-6 


5-« 


160.0 


47.0 


9.8 


293 -3 


39 


3 


2.0 


33-3 


72.0 


6.0 


166.6 


46.5 


10. 


300.0 


39 





2.2 


40.0 


69.2 


6.2 


173-3 


46.0 


10.25 


308.3 


38 


6 


2.4 


46.6 


66.7 


6.4 


180.0 


45-5 


10.50 


316.6 


38 


5 


2.6 


53-3 


61.9 


6.6 


186.6 


45 -° 


i°-75 


325-° 


38 





2.8 


60.0 


62.4 


6.8 


193-3 


44-5 


11.00 


333-3 


37 


9 


3-° 


66.6 


60.7 


7.0 


200.0 


44.0 


11.25 


341.6 


37 


7 


3? 


73-3 


59-i 


7.2 


206.6 


43 - 6 


11.50 


350-0 


37 


4 


3-4 


80.0 


57 ■« 


7-4 


213-3 


43-i 


n-75 


353-3 


37 


1 


3-6 


86.6 


5 6 -4 


7.6 


220.0 


42.8 


12 .00 


366.6 


36 


9 


3. » 


93-3 


55-2 


7.8 


226. 6 


42.4 


12.25 


375 -o 


30 


7 


4.0 


100. 


54-i 


8.0 


233-3 


42.0 


12.50 


3 8 3-3 


3b 


4 


4.2 


106. 6 


53-i 


8.2 


240.0 


41.7 


12.75 


391.6 


36 


2 


4.4 


U3-3 


5 2 -i 


8.4 


246.6 


41.4 


13.0 


400.0 


36 





4.6 


120.0 


5i-3 


8.6 


253-3 


41. 1 


14.0 


433-3 


35 


2 


4.8 


126.6 


5°-5 


8.8 


260.0 


40.8 


15.0 


466.6 


34 


s 


5-o 


133-3 


49-7 


9.0 


266.6 


40.5 


16.0 


500.0 


33 


8 



* This table reveals the limit of efficiency when air is applied without utiliz- 
ing any of its expansive energy. 

The column headed r gives the ratio of compression, while that headed h gives 
the water head equivalent to a pressure given by the ratio r on the assumption 
that one atmosphere is a pressure of 14.5 lb. per square inch or a water head of 
33.3 ft., this being more nearly the average condition than 14.7, which is so 
commonly taken. 

It should be understood that this efficiency cannot be reached in practice — 
it being reduced by friction of air and machinery and by clearance in any form 
of engine. 



TABLES 



139 



Table VII. — Efficiency of Direct Hydraulic Air Compressors 
Formula 33, Art. 32, viz. E = — — 



Water Head 


Gage Pressure 


Absolute Pres- 


Atmospheres 


Efficiency, 
E 






sure 


— r 


0.0 


0.0 


14-5 


1 


1 .00 


33-3 


14-5 


29.0 


2 


.69 


66.6 


29.0 


43-5 


3 


•55 


100. 


43-5 


58.0 


4 


.46 


133-3 


58.0 


7 2 -5 


5 


.40 


166.6 


72-5 


87.0 


6 


•36 


200.0 


87.0 


101.5 


7 


•33 


233-3 


101.5 


116. 


8 


•3° 


266.0 


116. 


i3o-5 


9 


.28 


300.0 


130-5 


145-° 


10 


.26 



Table VIII. — Coefficient "c" for Various Heads and Diameters 



d" 


i=i" 


i=2" 


2 = 3" 


•=4 ,# 


»=s" 


5 
TS 


0.603 


0.606 


0.610 


0.613 


0.616 


1 
2 


0.602 


0.605 


0.608 


0.610 


0.613 


I 


0.601 


0.603 


0.605 


0. 606 


0.607 


li 


0.601 


0.601 


0. 602 


0.603 


0. 603 


2 


0.600 


0.600 


0.600 


0.600 


0. 600 


,1 


0-599 


0-599- 


0-599 


0.598 


0.598 


3 


599 


0.598 


o-597 


0.596 


596 


& 


599 


o-597 


0.596 


0-595 


o.594 


4 


0.598 


o-597 


o-595 


o-594 


o-S93 


4^ 


598 


0.596 


0.596 


o-593 


0.592 



Tables VIII and Villa give the experimental coefficients for orifices for deter- 
mining the weight of air passing by formula : 



For round orifices 



Weight (Q) = c 0.1639 d 2 \ 7 p 



4 



4 



For rectangular orifices Weight (Q) = c 2.413 a 

Q = Weight of air passing in pounds per second. 
c = Experimental coefficient. 
d = Diameter of orifice in inches. 

i = Difference of pressure inside and outside of orifice in inches of water. 
t = Absolute temperature of air back of orifice. 
a = Area of rectangular orifice in square feet. 

p = Absolute pressure back of orifice in pounds per square inch = atmospheric 
pressure + 0.036 i. 



140 



COMPRESSED AIR 
Table Villa 









Coefficients "c" for Large Orifices 




Water 


McGill 
Coeffi- 




















Gage 


cient 




Round 




Square 


] Inches 


Orifice 
3H" 












1 
















30" 

1 


24" 


18" 


24"X24" 


i8"Xi8" 


i8"X30" 


I 


•599 


.604 


•599 


•597 


.607 


.598 


.602 


2 


•597 


.602 


•597 


• 596 


.605 


.596 


.600 


3 


•596 


.601 


•596 


•594 


.604 


•595 


•599 


4 


•597 


.600 


•595 


•593 


• 603 


• 694 


•598 


5 


•594 


• 599 


•594 


•592 


.601 


•593 


•597 



From Table IX can be readily found friction losses in air pipes as computed 
by the author's formula: Art. (26), viz., 

The table conforms to values of c taken from the curve A, B, Plate II, using the 
National Tube Works standard for actual diameters as shown here. 

National Tube Works Standard With Coefficients for Formula (26) 



Nominal diameters. 
Actual diameters . . 
Coefficient 



Yi 


H: 


I 


iH 


iM 


i« 


2 


2M 


1 

3 


.666 


.824 


I.048 


1-380 


1. 610 


1.820 


2.067 


2.468 


3.067 


. 170 


. 107 


•099 


.091 


.087 


.084 


.081 


0.765 


• 0715 



Nominal diameters . . 

Actual diameters 

Coefficient 



3^ 

3-548 
.0685 



4 
4.067 
.0660 



4M 
4.508 
•0635 



5 
5 -045 
.0615 



6 
6.065 
.0580 



7.981 
■0535 



10 
10.018 
.0500 



12 
12.00 

.0480 



TABLES 

Table IX. — Friction in Air Pipes 



141 





Divide the Number Corresponding to the Diameter and Volume by the 
Ratio of Compression. The Result is the Loss in Pounds per 
Square Inch in 1000 feet of Pipe 


Cubic Feet 

Free Air 

per Minute 










Nominal Diameter in Inches 


H 


% 


1 


iH 


iVt 1% 


2 


2V1 


3 


5 


12.7 


1.2 
















IO 


5°-7 


7.8 


2. 2 














IS 


114. 1 


17.6 


4-9 














20 




30-4 


8.7 


2.0 












25 




50.0 


13.6 


3-2 












3° 




70.4 


19.6 


4-5 












35 




95-9 


26.6 


6.2 


2.7 










40 




125.3 


34-8 


8.1 


3-6 


1.9 








45 






44.0 


10. 2 


4-5 


2.4 








5° 






54-4 


12.0 


5-6 


2.9 








60 






78.3 


18.2 


8.0 


4.2 


2. 2 






70 






106.6 


24.7 


10.9 


5-7 


2.9 






80 






139.2 


32.3 


14-3 


7-5 


3-8 






90 








40.9 


18. 1 


9-5 


4.8 






100 








50.5 


22.3 


11. 7 


6.0 






no 








61. 1 


27.0 


14. 1 


7.2 


2.8 




120 








72.7 


32.2 


16.8 


8.6 


3-3 




130 








85-3 


37-8 


19.7 


10. 1 


3-9 




140 






. . . -r 


98.9 


43-8 


22.9 


11. 7 


4.6 




150 








H3-6 


5°-3 


26.3 


13-4 


5-2 




160 








129.3 


57-2 


29.9 


15-3 


5-9 




170 










64.6 


33-7 


17.6 


6.7 




180 






.... 




72.6 


37-9 


19.4 


7-5 




190 










80.7 


42.2 


21.5 


8.4 


2.1 


200 










89.4 


46.7 


23-9 


9-3 


2.9 


220 










108.2 


56.5 


28.9 


n-3 


3-5 


240 










128.7 


67-3 


34-4 


13-4 


4.2 


260 












79.0 


4°-3 


15-7 


4.9 


280 












91.6 


46.8 


18.2 


5-7 


300 












105. 1 


53-7 


20.9 


6.6 



142 



COMPRESSED AIR 
Table IX. — (Continued) 







Nominal Diameter 


in Inches 




2 


2\i 


3 


3J-S 


4 


4 ! /i 


s 


6 


8 


320 


61. 1 


23-8 


7-5 


3-5 












340 


69.0 


26.8 


8.4 


3-9 












360 


77-3 


30.1 


9-5 


4-4 












380 


86.1 


33-5 


10.5 


5-9 












400 


94-7 


37-i 


11. 7 


5-4 


2.7 










420 


105.2 


40.9 


12.9 


6.0 


3-i 










440 


115 -5 


44.9 


14. 1 


6.6 


3-4 










460 


125.6 


48.8 


i5-4 


7-1 


3-7 










480 




53-4 


16.8 


7.8 


4.0 










500 




58-0 


18.3 


8-5 


4-3 










525 




64.2 


20. 2 


9-4 


4.8 


2.6 








550 




70. 2 


22. 1 


10.2 


5-2 


2.9 








575 




76.7 


24.2 


11. 2 


5-7 


3-i 








600 




83-5 


26.3 


12. 2 


6.2 


3-4 








625 




92.7 


28. s 


13.2 


6.8 


3-7 








650 




98.0 


39-9 


14-3 


7-3 


4.0 








675 




105.7 


33-3 


15-4 


7-9 


4-3 








700 




II3-7 


35-8 


16.6 


8-5 


4.6 








75o 




130.5 


41. 1 


19.0 


9-7 


5-3 


2.9 






800 






46.7 


21.7 


11. 1 


6.1 


3-3 






850 






52.8 


24.4 


12.5 


6.8 


3-8 






900 






59-i 


27.4 


14.0 


7-7 


4.2 






95° 






65-9 


30.5 


15-7 


8.6 


4-7 






1000 






73 -o 


33-8 


17-3 


9-5 


5-2 






1050 






80.5 


37-3 


19. 1 


10.4 


5-8 






1 100 






88.4 


40.9 


21 .0 


"•5 


6-3 


2.4 




1150 






96.6 


44-7 


22.9 


12.5 


6.9 


2.6 




1200 






105.2 


48.8 


25.0 


13-7 


7-5 


2.8 




1300 






123.4 


57-2 


29-3 


16.0 


8.8 


3-3 




1400 




.... 




66.3 


33-9 


18.6 


10.2 


3-8 




1500 








76.1 


39-o 


21.3 


11. 8 


4-4 




1600 








86.6 


44-3 


24.2 


13-4 


5-i 




1700 








97-8 


5o.i 


27.4 


i5-i 


5-7 




1800 


.... 






110.0 


56.1 


30.7 


16.9 


6.4 





TABLES 

Table IX. — (Continued) 



143 









N 


ominal Diametei 


in Inches 


4 


4W 


S 


6 


8 


10 


12 






1900 


62.7 


34-2 


18.9 


7-i 












2000 


69-3 


37-9 


21.3 


7.8 












2100 


76.4 


40.8 


23.O 


8.7 


2.0 










2200 


83.6 


45-8 


25-3 


9-5 


2 . 2 










2300 


91 .6 


50.1 


27.6 


10.4 


2-4 










2400 


99.8 


54-6 


30.1 


n-3 


2.6 










2500 


108.3 


59-2 


32.6 


12.3 


2.9 










2600 


117. 2 


64.0 


35-3 


13-3 


3-1 










2700 




69.1 


38.1 


14-3 


3-3 










2800 




74-3 


41.0 


15-4 


3-6 










2900 




79.8 


43-9 


16.5 


3-9 










3000 




85.2 


47 -o 


17.7 


4.1 










3200 




97.1 


53-5 


20. 1 


4-7 










3400 




109.5 


60.4 


22 . 7 


5-3 










3600 




122 .8 


67.7 


25-4 


5-6 










3800 






75-5 


28.4 


6.6 










4000 






83.6 


3i-4 


7-3 










4200 






92 . 1 


34-6 


8.1 










4400 






101 . 2 


38.1 


8.9 










4600 






no. 5 


4i-5 


9-7 


2.9 








4800 






120.4 


45-2 


10.5 


3-2 








5000 








49.1 


ii-S 


3-4 








5250 








54-i 


12.6 


3-8 








55°o 








59-4 


13-9 


4-2 








575° 








64.9 


15-2 


4-6 








6000 








70.7 


16.5 


5-o 








6500 








82.9 


19.8 


5-9 


2-3 






7000 








96.2 


22.5 


6.8 


2.6 






7500 








no. 5 


25-8 


7-8 


3-o 






8000 








125-7 


29.4 


8.8 


3-6 






9000 










37-2 


n. 2 


4-4 






1 0000 










45-9 


138 


5-4 






1 1000 










55-5 


16. 7 


6-5 






12000 










66.1 


19-8 


7-7 






13000 










77-5 


2 3-3 


9.0 






14000 










89.9 


27.0 


10.5 






15000 










103.2 


3i-o 


12.0 






16000 










117. 7 


35-3 


13-7 






18000 










148.7 


44-6 


17.4 






20000 












55-o 


21 .4 






22000 












66.9 


26.0 






24000 












79-3 


30.1 






26000 












93-3 


36.3 






28000 












108.0 


42.1 






30000 






.:.. 


_ 




123.9 


48.2 







144 



COMPRESSED AIR 



Table X. — Table of Contents of Pipes in Cubic Feet and in U. S. Gallon 







For 1 Foot 


in Length 






For 1 Foot 


— — — — 
in Length 


















Diam. 


in Deci- 


Cubic Feet. 


Gallons of 


Diam. 


in Deci- 


Cubic Feet. 


Gallons of 


in 
Inches 


mals of 
a Foot 


Also Area 

in Square 

Feet 


23 1 Cubic 
Inches 


in 
Inches 


mals of 
a Foot 


Also Area 

in Square 

Feet 


23 1 Cubic 
Inches 


1 

4 


.0208 


.0003 


.0026 


11. 


.9167 


.6600 


4-937 


h 


.0260 


.0005 


.0040 


1 
4 


•9375 


.6903 


5-163 


i 


•0313 


.0008 


.0057 


i 


•9583 


.7213 


5-395 


A 


•0365 


.0010 


.0078 


a 

4 


•9792 


•7530 


5-633 


i 


.0417 


" .0014 


.0102 


12. 


1 Foot 


•7854 


5.876 


A 


.0469 


.0017 


.0129 


1 


1.042 


•8523 


6-375 


f 


.0521 


.0021 


.0159 


13- 


1.083 


.9218 


6.895 


ii 

TK 


•o573 


.0026 


.0193 


\ 


1-125 


.9940 


7-435 


a 

4 


.0625 


.0031 


.0230 


14. 


1. 167 


I.069 


7-997 


tt 


.0677 


.0036 


.0270 


\ 


1.208 


1. 147 


8-578 


7 

s 


.0729 


.0042 


.0312 


15- 


1.250 


I.227 


9.180 


15 
16 


.0781 


.0048 


•°359 


1 

2 


1.292 


1,310 


9.801 


I . 


•0833 


•OOS5 


.0408 


16. 


1-333 


I.396 


10.44 


1 
1 


.1042 


.0085 


.0638 


1 

2 


1-375 


I-485 


11. 11 


1 

5 


.1250 


.0123 


.0918 


i7- 


1-417 


1-576 


11.79 


2. 

4 


.1458 


.0168 


.1250 


1 
2 


1.458 


I.670 


12.50 


2. 


.1667 


.0218 


.1632 


18. 


1.500 


I.767 


13.22 


1 
4 


•i875 


.0276 


.2066 


1 
2 


1.542 


I.867 


13-97 


J 


.2083 


.0341 


•2550 


19. 


1-583 


I.969 


J 4-73 


4 


.2292 


.0413 


•3085 


1 

2 


1.625 


2.074 


I5-52 


3- 


.2500 


.0491 


•3673 


20. 


1.666 


2.182 


16.32 


i 

4 


.2708 


.0576 


• 431° 


1 

2 


1.708 


2.292 


I7-I5 


1 

2 


.2917 


.0668 


.4998 


21. 


1-75° 


2.405 


17.99 


3 
4 


•3125 


.0767 


•5738 


1 

2 


1.792 


2.521 


18.86 


4- 


•3333 


.0873 


.6528 


22. 


i-833 


2.640 


19-75 


i 

4 


•3542 


.0985 


•737o 


1 

2 


i-875 


2.761 


20.65 


1 
5 


•375° 


.1105 


.8263 


2 3- 


i-9i7 


2.885 


22.58 


3 
4 


•3958 


.1231 


.9205 


1 
2 


1.958 


3.012 


21-53 


5- 


.4167 


•I3 6 4 


1.020 


24. 


2.000 


3.142 


23-50 


i 

4 


•4375 


•1503 


1. 124 


2 5- 


2.083 


3-409 


25-5o 


1 
2 


•4583 


.1650 


1-234 


26. 


2. 166 


3.687 


27-58 


I 


.4792 


.1803 


1-349 


27. 


2.250 


3-976 


29.74 


6. 


.5000 


.1963 


1.469 


28. 


2-333 


4.276 


3i-99 


i 

4 


.5208 


.2130 


1-594 


29. 


2.416 


4-587 


34-31 


1 

5 


•5417 


•2305 


1.724 


3°- 


2.500 


4.909 


36.72 


1 


■5625 


.2485 


j. 859 


3i- 


2.583 


5-241 


39.21 


7 


•5833 


.2673 


1.999 


32. 


2.666 


5-585 


41.78 


i 

4 


.6042 


.2868 


2.144 


33- 


2.750 


5-940 


44-43 


1 


.6250 


.3068 


2.295 


34 


2-833 


6-305 


47-17 


3 
4 


.6458 


•3275 


2.450 


35- 


2.916 


6.681 


49.98 


8. 


.6667 


•349° 


2. 61 1 


36. 


3.000 


7.069 


52.88 


i 

4 


.6875 


•3713 


2.777 


37- 


3-083 


7.468 


55-86 


a 


.7083 


•3940- 


2.948 


38. 


3.166 


7.876 


58.92 


a 

•4 


.7292 


•417S 


3-125 


39- 


3-250 


8.296 


62 . 06 


9- 


.7500 


.4418 


3-3°5 


40. 


3-333 


8.728 


65.29 


i 

4 


.7708 


.4668 


3-492 


41. 


3.416 


9.168 


68.58 


1 

2 


.7917 


•4923 


3.682 


42. 


3-5oo 


9. 620 


71.96 


t 


• 8125 


•5185 


3-879 


43- 


3-583 


10.084 


75-43 


IO. 


•8333 


•5455 


4.081 


44- 


3.666 


10.560 


79.00 


1 
4 


.8542 


•573° 


4.286 


45- 


3-750 


I I . 044 


82.62 


i 


.8750 


.6013 


4.498 


46. 


3-833 


II.540 


86.32 


a 

4 


.8958 


• 6303 


4.714 


47- 


3.916 


12.048 


90. 12 










48. 


4.000 


12.566 


94.02 



Table XI. — Cylindrical Vessels, Tanks, Cisterns, Etc. 

Diameter in Feet and Inches, Area in Square Feet, and U. S. Gallons Capacity 

for One Foot in Depth 

i . i cu. ft. 

i gal. = 231 cu. in. = - — = 0.13368 cu. ft. 

7.4805 



Diam. 


Area 


Gals. 


Diam. 


Area 


Gals. 


Diam. 


Area 


Gals. 


Ft. In. 


Sq. Ft. 


1 Ft. 
Depth 


Ft. 


In. 


Sq. Ft. 


1 Ft. 
Depth 

172.38 


Ft. 


In. 


Sq. Ft. 


1 Ft. 
Depth 


I 


•785 


5-89 


5 


5 


23.04 


17 


6 


240.53 


1799-3 


I I 




922 


6.89 


5 


6 


23.76 


177.72 


17 


9 


247-45 


1851.1 


I 2 


I 


069 


8.00 


5 


7 


24.48 


I83-I5 


18 




254-47 


1903.6 


1 3 


I 


227 


9.18 


5 


8 


25.22 


188.66 


18 


3 


261.59 


1956.8 


1 4 


I 


396 


IO.44 


5 


9 


2 5-97 


194.25 


18 


6 


268.80 


2010.8 


1 5 


I 


576 


II.79 


5 


10 


26.73 


199.92 


18 


9 


276.12 


2065.5 


1 6 


I 


767 


13.22 


5 


11 


27.49 


205 . 67 


19 




2 83-53 


2120.9 


1 7 


I 


969 


14-73 


6 




28.27 


211. 51 


19 


3 


291.04 


2177. 1 


1 8 


2 


182 


16.32 


6 


3 


30.68 


229.50 


19 


6 


298.65 


2234.O 


1 9 


2 


405 


17.99 


6 


6 


33.18 


248.23 


19 


9 


306.35 


2291 .7 


1 10 


2 


640 


19-75 


6 


9 


35.78 


267.69 


20 




314.16 


2.350-I 


in 


2 


885 


21.58 


7 




38.48 


287.88 


20 


3 


322.06 


2409. 2 


2 


3 


142 


2 3-5° 


7 


3 


41.28 


308.81 


20 


6 


330.06 


2469. 1 


2 1 


3 


409 


2 5-5° 


7 


6 


44.18 


33 .48 


20 


9 


338.16 


2529.6 


2 2 


3 


687 


27-58 


7 


9 


47-17 


352.88 


21 




346.36 


2591.0 


2 3 


3 


976 


29.74 


8 




50.27 


376.01 


21 


3 


354-66 


2653.0 


2 4 


4 


276 


31-99 


8 


3 


53-46 


399.88 


21 


6 


363-05 


2715.8 


2 5 


4 


587 


34-3i 


8 


6 


56-75 


424.48 


21 


9 


371-54 


2779.3 


2 6 


4 


909 


36.72 


8 


9 


60.13 


449.82 


22 




380.13 


2843.6 


2 7 


5 


241 


39.21 


9 




63.62 


475-89 


22 


3 


388.82 


2908.6 


2 8 


5 


585 


41.78 


9 


3 


67.20 


502.70 


22 


6 


397- 61 


2974-3 


2 9 


5 


940 


44-43 


9 


6 


70.88 


53°- 2 4 


22 


9 


406 . 49 


3040.8 


2 10 


6 


3°5 


47-16 


9 


9 


74.66 


558.51 


2 3 




415.48 


3108.0 


2 11 


6 


681 


49.98 


10 




78-54 


5 8 7-52 


2 3 


3 


424.56 


3I75-9 


3 


7 


069 


52.88 


10 


3 


82.52 


617.26 


2 3 


6 


433-74 


3244.6 


3 1 


7 


467 


55-86 


10 


6 


86.59 


647.74 


23 


9 


443- 01 


33I4-0 


3 2 


7 


876 


58.92 


10 


9 


90.76 


678.95 


24 




45 2 -39 


3384-I 


3- 3 


8 


296 


62.06 


11 




95-o3 


710.90 


24 


3 


461.86 


3455 -o 


3 4 


8 


727 


65.28 


11 


3 


99.40 


743-58 


24 


6 


471-44 


3526.6 


3 5 


9 


168 


68.58 


11 


6 


103.87 


776.99 


24 


9 


481. 11 


3598.9 


3 6 


9 


621 


71.97 


11 


9 


108.43 


811. 14 


25 




490.87 


3672.0 


3 7 


10 


085 


75-44 


12 




113-1° 


846.03 


2 5 


3 


500.74 


3745-8 


3 8 


10 


559 


78.99 


12 


3 


117.86 


881 . 65 


2 5 


6 


510.71 


3820.3 


3 9 


11 


045 


82.62 


12 


6 


122.72 


918.00 


2 5 


9 


520.77 


3895-6 


3 10 


11 


541 


86.33 


12 


9. 


127.68 


955-09 


26 




530-93 


3971.6 


3 11 


12 


048 


90.13 


13 




i3 2 -73 


992.91 


26 


3 


54.1 . 19 


4048.4 


4 


12 


566 


94.00 


13 


3 


137-89 


1031-5 


26 


6 


551-55 


4125.9 


4 1 


13 


095 


97.96 


13 


6 


I43-I4 


1070.8 


26 


9 


562.00 


4204.1 


4 2 


13 


635 


102.00 


13 


9 


148.49 


1 1 10. 8 


27 




572.56 


4283.0 


4 3 


14 


186 


106.12 


14 




153-94 


ii5i.5 


27 


3 


583-21 


4362.7 


4 4 


14 


748 


110.32 


14 


3 


159.48 


1 193.0 


27 


6 


593-96 


4443 • 1 


4 5 


iS 


321 


114. 61 


14 


6 


165-13 


1235-3 


27 


9 


604.81 


4524.3 


4 6 


15 


90 


118.97 


14 


9 


170.87 


1278. 2 


28 




6i5-75 


4606.2 


4 7 


16 


5° 


123.42 


i5 




176.71 


1321-9 


28 


3 


626.80 


4688.8 


4 8 


17 


10 


i 2 7-95 


i5 


3 


182.65 


1366.4 


28 


6 


637-94 


4772.1 


4 9 


17 


72 


i3 2 -5 6 


i5 


6 


188.69 


I4H-5 


28 


9 


649. 18 


4856.2 


4 10 


18 


35 


i37- 2 5 


i5 


9 


194.83 


1457-4 


29 




660.52 


4941.0 


4 11 


18 


99 


142.02 


16 




201.06 


1504-1 


29 


3 


671:96 


5026.6 


5 


19 


63 


146.88 


16 


3 


207.39 


I55I-4 


29 


6 


683 . 49 


5112-9 


5 1 


20 


29 


151.82 


16 


6 


213.82 


1599-5 


29 


9 


695 • 13 


5I99.9 


5 2 


20 


97 


156.83 


16 


9 


220.35 


1648.4 


30 




706.86 


5287.7 


5 3 


21 


65 


161.93 


17 




226.98 


1697.9 










S 4 


22 


34 167.12 


17 


3 


233 • 7J 


1748.2 











10 



145 



146 



COMPRESSED AIR 



Table XII. — Standard Dimensions of Wrought-iron Welded Pipe 
(National Tube Works) 



Nominal 


Actual 


Actual 










Inside 


Outside 


Inside 


Internal Area 


External Area 


Diameter 


Diameter 


Diameter 










Ins. 


Ins. 


Ins. 


Sq. In. 


Sq. Ft. 


Sq. In. 


Sq. Ft. 


1 


•405 


.270 


•057 


.0004 


.1288 


. 0009 


1 


•540 


•3 6 4 


.104 


.0007 


.2290 


.0016 


t 


•675 


•493 


.191 


.0013 


•3578 


.0025 


1 


.840 


.622 


•3°4 


.0021 


•554 


.0038 


1 


1.050 


.824 


•533 


.0037 


.866 


.0060 


I 


I-3IS 


1.048 


.861 


.0060 


i-358 


.0094 


ll 


1.660 


1.380 


1.496 


.0104 


2.164 


.0150 


Ij 


I.900 


1. 610 


2.036 


.0141 


2.835 


.0197 


2 


2-375 


2.067 


3-356 


•0233 


4-43° 


.0308 


?1 


2.875 


2.468 


4.780 


•0332 


6.492 


.0451 


3 


3-5°° 


3.067 


7-383 


•05I3 


9.621 


.0668 


3l 


4.000 


3-548 


9.887 


.0689 


12.566 


.0875 


4 


4.500 


4.026 


12.730 


.0884 


15.904 


.1104 


4l 


5.000 


4.508 


15.961 


.1108 


I9-635 


.1364 


5 


5-563 


5- 045 


19.986 


.1388 


24.301 


.1688 


6 


6.625 


6.065 


28.890 


.2006 


34-472 


•2394 


7 


7.625 


7.023 


38-738 


.2690 


45 • 664 


•3171 


8 


8.625 


7.981 


50.027 


•3474 


58.426 


•4057 


9 


9.625 


8-937 


62 . 730 


•4356 


72.760 


•5053 


IO 


io-75 


10.018 


78.823 


•5474 


90.763 


•6303 


ii 


n-75 


1 1 . 000 


95-°33 


.6600 


108.434 


•7530 


-12 


12.75 


12.000 


113.098 


•7854 


127.677 


.8867 


13 


14 


13-25 


I37-887 


•9577 


I53-938 


I.0690 


■ 14 


15 


14.25 


I59-485 


1-1075 


176.715 


I.2272 


15 


16 


I5-25 


182.665 


1.2685 


201.062 


I-3963 


17 


18 


17-25 


239.706 


1.6229 


254.470 


I. 7671 


19 


20 


19.25 


291 .040 


2 .0211 


3I4-I59 


2.1817 


21 


22 


21.25 


354-657 


2.4629 


380.134 


2.6398 


23 


24 


2 3-25 


424-55 8 


2.9483 


452-39° 


3-I4I6 



TABLES 



147 





Table XIII 


— Hyperbolic Logarithms 




N 


Loga- 


N 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 


1 


ithm. 




ithni. 




rithm. 


I.OI 


.00995 


i-57 


.45108 


2.13 


.75612 


2.69 


.98954 


1. 02 


.01980 


1.58 


•45742 


2.14 


. 76081 


2.70 


.99325 


1.03 


.02956 


1-59 


•46373 


2.15 


•76547 


2.71 


•99695 


1.04 


.03922 


1.60 


.47000 


2.l6 


.77011 


2.72 


1.00063 


1.05 


.04879 


1.61 


47623 


2.17 


77473 


2-73 


1 . 00430 


1.06 


.05827 


1.62 


48243 


2.18 


77932 


2.74 


1 .00796 


1.07 


.06766 


1.63 


48858 


2.19 


78390 


2-75 


1. 01 1 60 


1.08 


.07696 


1.64 


49470 


2.20 


78846 


2.76 


1. 01523 


1.09 


.08618 


1.65 


50078 


2.21 


79299 


2.77 


1.018S5 


1.10 


•09531 


1.66 


50681 


2.22 


79751 


2.78 


1.02245 


1. 11 


.10436 


1.67 


51282 


2.23 


80200 


2.79 


1.02604 


1. 12 


•H333 


1.68 


5i 8 79 


2.24 


80648 


2.80 


1.02962 


r.13 


. 12222 


1.69 


52473 


2.25 


81093 


2.81 


1. 03318 


1. 14 


■ 1 3^°3 


1.70 


53063 


2.26 


81536 


2.82 


1.03674 


1. 15 


•13977 


1.71 


53649 


2.27 


81978 


2.83 


1.04028 


1. 16 


. 14842 


1.72 


54232 


2.28 


82418 


2.84 


1.04380 


1.17 


.15700 


i-73 


54812 


2.29 


82855 


2.85 


1.04732 


1. 18 


•i°55i 


1.74 


55389 


2.30 


83291 


2.86 


1.05082 


1. 19 


•17395 


i-75 


55962 


2.31 


83725 


2.87 


1-05431 


1.20 


. 18232 


1.76 


56531 


2.32 


84157 


2.88 


1.05779 


1. 21 


. 19062 


1.77 


57098 


2-33 


84587 


2.89 


1. 06126 


1.22 


.19885 


1.78 


5766i 


2-34 


8 5°i5 


2.90 


1. 06471 


1.23 


.20701 


1.79 


58222 


2-35 


85442 


2.91 


1. 06815 


1.24 


.21511 


1.80 


58779 


2.36 


85866 


2.92 


1. 07158 


1.25 


.22314 


1. 81 


59333 


2-37 


86289 


2-93 


1.07500 


1.26 


.23111 


1.82 


59884 


2.38 


86710 


2.94 


1. 07841 


1.27 


.23902 


1.83 


60432 


2-39 


87129 


2-95 


1.08181 


1.28 


.24686 


1.84 


60977 


2.4O 


87547 


2.96 


1. 085 19 


1.29 


•25464 


1.85 


61519 


2.41 


87963 


2-97 


1.08856 


1.30 


.26236 


1.86 


62058 


2.42 


88377 


2.98 


1 .09192 


1 -31 


.27003 


1.87 


62594 


2-43 


88789 


2.99 


1.09527 


1.32 


.27763 


1.88 


63127 


2.44 


89200 


3.00 


1. 09861 


i-33 


.28518 


1.89 


63658 


2-45 


89609 


3.01 


1. 10194 


1-34 


.29267 


1.90 


64185 


2.46 


90016 


3.02 


1. 10526 


1-35 


.30010 


1. 91 


64710 


2-47 


90422 


3-03 


1. 10856 


1.36 


.30748 


1.92 


65233 


2.48 


90826 


3 -04 


1.11186 


i-37 


.31481 


1-93 


65752 


2.49 


91228 


3 -05 


1.. 11514 


1.38 


.32208 


1.94 


66269 


2.50 


91629 


3-o6 


1. 11841 


1-39 


.32930 


1-95 


66783 


2.51 


92028 


3-07 


1.12168 


1.40 


•33647 


1.96 . 


67294 


2.52 


92426 


3-o8 


1. 12493 


1. 41 


•34359 


1.97 


67803 


2.53 


92822 


3-09 


1.12817 


1.42 


.35066 


1.98 


68310 


2.54 


93216 


3-io 


1.13140 


1-43 


•35767 


1.99 


68813 


2.55 


93609 


3-ii 


1. 13462 


1.44 


• 36464 


2.00 


69315 


2.56 


94001 


3.12 


I-I3783 


i-45 


•37156 


2.01 


69813 


2.57 


94391 


3-13 


1.14103 


1.46 


•37 8 44 


2.02 


70310 


2.58 


94779 


3-14 


1. 14422 


1.47 


.38526 


2.03 


70804 


2-59 


95166 


3-15 


1 . 14740 


1.48 


.39204 


2.04 


71295 


2.60 


95551 


3.16 


I-I5057 


1.49 


•39878 


2.05 


71784 


2.6l 


95935 


3-17 


1 -15373 


1.50 


•40547 


2.06 


72271 


2.62 


96317 


3.18 


1. 15688 


1 -5i 


.41211 


2.07 


72755 


2.63 


96698 


3-19 


1 . 16002 


1.52 


.41871 


2.08 


73237 


2.64 


97078 


3.20 


1-16315 


1-53 


.42527 


2.09 


73716 


2.65 


97454 


3-21 


1 . 16627 


i-54 


•43178 


2.10 


74194 


2.66 


97833 


3-22 


1. 16938 


i-55 


•43825 


2. 11 


74669 


2.67 


98208 


3 23 


1. 17248 


.1.56 


• 4446q 


2.12 


75142 


2.68 


98582 


3.24 


1 -17557 



148 



COMPRESSED AIR 





Table XIII. Continued — 


-Hyperbolic Logarithms 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 




rithm. 




rithm. 


3-25 


1. 17865 


3.8l 


I-33763 


4-37 


1.47476 


4-93 


1-59534 


3 26 


1.18173 


3.82 


1-34025 


4-38 


I-47705 


4.94 


!• 59737 


3-27 


1. 18479 


3.83 


1.34286 


4-39 


1-47933 


4-95 


1-59939 


3.28 


1. 18784 


3-84 


1-34547 


4.40 


1. 48160 


4.96 


1.60141 


3-29 


1 19089 


3.85 


1.34807 


4.41 


1.48387 


4-97 


1 . 60342 


3.30 


1 . 19392 


3-86 


i-35o67 


4.42 


1. 48614 


4.98 


1.60543 


3-3i 


1 . 19695 


3.87 


1 -353 2 5 


4-43 


1 . 48840 


4.99 


1 . 60744 


3.32 


1 . 19996 


3-88 


i-355 8 4 


4.44 


1.49065- 


5-oo 


1 . 60944 


3-33 


1.20297 


3.89 


i-35 8 4r 


4-45 


1.49290 


5.01 


1.61144 


3-34 


1.20597 


3-90 


1 . 36098 


4.46 


J -495i5 


5-02 


1 61343 


3-35 


1.20896 


3-91 


I-36354 


4-47 


1-49739 


5.03 


1. 61542 


3-36 


1.21194 


3 -92 


1.36609 


4.48 


1 . 49962 


5-04 


1.61741 


3-37 


1 .21491 


3-93 


1.36864 


4.49 


1. 50185 


5 -05 


1. 61939 


3.38 


1. 21788 


3-94 


1.37118 


4-5o 


1 . 50408 


5.06 


1. 62137 


3-39 


1.22083 


3-95 


I-3737I 


4.51 


1.50630 


5-07 


1 . 62334 


3 40 


1.22378 


3 96 


1.37624 


4-52 


1-50851 


5.08 


1. 62531 


3-4i 


1. 22671 


3-97 


I-37877 


4-53 


1. 51072 


5-09 


1.62728 


3-42 


1.22964 


3.98 


1. 38128 


4-54 


!-5i293 


5.10 


1 . 62924 


3-43 


1.23256 


3-99 


I-38379 


4-55 


r-5i5i3 


5.11 


1 .63120 


3-44 


1-23547 


4.00 


1.38629 


4-56 


I-5I732 


5-12 


1-63315 


3-45 


1-23837 


4.01 


1.38879 


4-57 


I-5I95 1 


5-i3 


1-63511 


346 


1. 24127 


4.02 


1. 39128 


4-58 


1. 52170 


5.14 


1-63705 


3-47 


1. 24415 


4-03 


1-39377 


4-59 


1.52388 


5.15 


1 . 63900 


3-48 


1 . 24703 


4.04 


1.39624 


4.60 


1.52606 


5.16 


1 . 64094 


3-49 


1 . 24990 


4-05 


1.39872 


4.61 


1.52823 


5-17 


1 . 64287 


3-50 


1.25276 


4.06 


1.40118 


4.62 


1 • 53039 


5.18 


1. 6448 1 


3.5i 


1.25562 


4.07 


1.40364 


463 


1-53256 


5-19 


1.64673 


3-52 


1.25846 


4.08 


1. 40610 


4.64 


i-5347i 


5-20 


1 . 64866 


3-53 


1. 26130 


4.09 


1.40854 


4-65 


1-53687 


5.21 


1.65058 


3-54 


1 .26412 


4.10 


1 .41099 


4.66 


1.53902 


5-22 


1.65250 


3-55 


1.26695 


4.11 


1. 41342 


4.67 


i-54ii6 


5-23 


1. 65441 


3-56 


1.26976 


4.12 


1-41585 


4.68 


1 • 5433o 


5.24 


.1.65632 


3-57 


1.27257 


4-13 


1. 41828 


4.69 


* -54543 


5-25 


1.65823 


3.58 


I-27536 


4.14 


1 .42070 


4.70 


1-54756 


5.26 


1. 66013 


3-59 


1. 27815 


4.15 


1-42311 


4.71 


1 . 54969 


5-27 


1 . 66203 


3-6o 


1 . 28093 


4.16 


I-4255 2 


4.72 


i-55i8i 


5.28 


1 . 66393 


3-6i 


1. 28371 


4.17 


1.42792 


4-73 


1-55393 


5-29 


1 . 66582 


3.62 


1 . 28647 


4.18 


1 -4303 1 


4-74 


1.55604 


5-30 


1. 66771 


3 63 


1 .28923 


4.19 


1.43270 


4-75 


i-558i4 


5.3i 


1 . 66959 


3.64 


1 .29198 


4.20 


1.43508 


4.76 


1 . 56025 


5-32 


1. 67147 


3.65 


1.29473 


4.21 


1.43746 


4-77 


1-56235 


5-33 


I-67335 


3-66 


1.29746 


4.22 


1.43984 


4.78 


1.56444 


5-34 


1.67523 


3.67 


1. 30019 


4.23 


1 .44220 


4-79 


1-56653 


5-35 


1. 67710 


3-68 


1 .30291 


4.24 


1.44456 


4.80 


1.56862 


5.36 


1 . 67896 


3 69 


i-305 6 3 


4-25 


1.44692 


4.81 


1.57070 


5-37 


1 . 680S3 


3-7o 


i-3° 8 33 


4.26 


1.44927 


4.82 


I-57277 


5.38 


1 . 68269 


3-71 


1.31103 


4.27 


1.45161 


4.83 


I-57485 


5-39 


1.68455 


3-72 


1. 31372 


4.28 


*-45395 


4.84 


1. 57691 


5-40 


1 . 68640 


3-73 


1.31641 


4.29 


1.45629 


4-85 


1.57898 


5.4i 


1.68825 


3-74 


1. 31909 


4-30 


1. 45861 


4.86 


1. 58104 


5-42 


1 .69010 


3-75 


1. 32176 


4-31 


1 . 46094 


4.87 


1 . 58309 


5-43 


1. 69194 


3.76 


1.32442 


4-32 


1 .46326 


4.88 


1-58515 


5-44 


1.69378 


3-77 


1.32707 


4-33 


I-46557 


4.89 


1. 58719 


5-45 


1.69562 


3.78 


1.32972 


4-34 


1.46787 


4.90 


1.58924 


5-46 


1 . 69745 


3-79 


I-33237 


4-35 


1 .47018 


4.91 


1. 59127 


5-47 


1 . 69928 


3.8o 


1 . 33500 


4-36 


1-47247 


4.Q2 


1 -59331 


5.48 


1 . 701 1 1 



TABLES 



149 





Table 


XIII. 


Continued.- 


—Hyperbolic Logarithms 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 




rithm. 




rithm. 


5-49 


1.70293 


6.05 


1 . 80006 


6.61 


1.88858 


7.17 


1. 96991 


5.50 


1 • 70475 


6.06 


1.80171 


6.62 


1. 89010 


7.18 


1. 97130 


5-5i 


1 .70656 


6.07 


1.80336 


6.63 


1. 89160 


7.19 


1 .97269 


5-52 


1.70838 


6.08 


1 . 80500 


6.64 


1-89311 


7.20 


1.97408 


5-53 


1 .71019 


6.09 


1.80665 


6.65 


1.89462 


7.21 


1 -97547 


5-54 


1.71199 


6.10 


1 . 80829 


6.66 


1. 89612 


7.22 


1.97685 


5-55 


1. 71380 


6.11 


1 . 80993 


6.67 


1.89762 


7-23 


1.97824 


5.56 


1. 71560 


6.12 


1.81156 


6.68 


1. 89912 


7.24 


1.97962 


5-57 


1. 71740 


6.13 


1-81319 


6.69 


1 .90061 


7-25 


1. 98100 


5-58 


1.71919 


6.14 


1. 81482 


6.70 


1. 902 1 1 


7.26 


1.98238 


5-59 


1.72098 


6.15 


1. 81 645 


6.71 


1.90360 


7.27 


1.98376 


5-6o 


1.72277 


6.16 


1. 81808 


6.72 


1.90509 


7.28 


1-98513 


5.61 


!• 7 2 455 


6.17 


1. 81970 


6-73 


1.90658 


7.29 


1.98650 


5-62 


I.72633 


6.18 


1 .82132 


6.74 


1 .90806 


7-30 


1.98787 


5-63 


I.72811 


6.19 


1 .82294 


6-75 


1 . 90954 


7-3i 


1.98924 


5.64 


I . 72988 


6.20 


1-82455 


6.76 


1 .91102 


7-32 


1. 9906 1 


5.65 


I. 73166 


6.21 


1. 82616 


6.77 


1. 91250 


7-33 


1. 99198 


5-66 


!■ 73342 


6.22 


1.82777 


6.78 


1. 91398 


7-34 


1-99334 


5-67 


I-735I9 


6.23 


1.82937 


6.79 


I-9I545 


7-35 


1.99470 


5.68 


I • 73695 


6.24 


1.83098 


6.80 


1. 91692 


7-36 


1.99606 


5-69 


I-7387I 


6.25 


1-83258 


6.81 


1. 91839 


7-37 


1.99742 


5.7o 


I . 74047 


6.26 


1. 83418 


6.82 


1 .91986 


7.38 


1.99877 


5.7i 


I .74222 


6.27 


I-83578 


6.83 


1 .92132 


7-39 


2.00013 


5.72 


1 • 74397 


6.28 


I-83737 


6.84 


1 .92279 


7.40 


2.00148 


5-73 


1.74572 


6.29 


1.83896 


6.85 


1.92425 


7.41 


2.00283 


5-74 


1.74746 


6.30 


1.84055 


6.86 


1. 92571 


7.42 


2.00418 


5-75 


1.74920 


6.31 


1. 84214 


6.87 


1. 92716 


7-43 


2.00553 


5.76 


1 . 75094 


6.32 


1.84372 


6.88 


1.92862 


7-44 


2.00687 


5-77 


1.75267 


6-33 


1.84530 


6.89 


1.93007 


7-45 


2 . 0082 1 


5.78 


1 . 75440 


6-34 


1 . 84688 


6.90 


I-93I52 


7.46 


2 .00956 


5-79 


1-75613 


6-35 


1.84845 


6.91 


1.93297 


7-47 


2.01089 


5.80 


1.75786 


6.36 


1.85003 


6.92 


1.93442 


7.48 


2.01223 


5.81 


i- 7595 s 


6-37 


1. 85160 


6-93 


1.93586 


7-49 


2-01357 


5-82 


1. 76130 


6.38 


I-853I7 


6.94 


1-9373° 


7-5o 


2.01490 


5.83 


1 .76302 


6-39 


1-85473 


6-95 


1.93874 


7-5i 


2.01624 


5-84 


1.76473 


6.40 


1.85630 


6.96 


1. 94018 


7-52 


2.01757 


5.85 


1 . 76644 


6.41 


1.85786 


6.97 


1 .94162 


7-53 


2.01890 


5-86 


1. 76815 


6.42 


1.85942 


6.98 


i-943°5 


7-54 


2.02022 


5-87 


1 . 76985 


6-43 


1.86097 


6-99 


1.94448 


7-55 


2.02155 


5-88 


1. 77156 


6.44 


1.86253 


7.00 


1 -94591 


7-56 


2 .02287 


5.89 


1.77326 


6-45 


1 . 86408 


7.01 


r-94734 


7-57 


2.02419 


5 -90 


!• 77495 


6.46 


1.86563 


7.02 


1.94876 


7.58 


2.02551 


5.9i 


1 . 77665 


6.47 


1. 86718 


7-03 


1. 95019 


7-59 


2 .02683 


5-92 


1 -77 8 34 


6.48 


1.86872 


7.04 


1.95161 


7.60 


2.02815 


5-93 


1 . 78002 


6.49 


1 .87026 


7-05 


i- 95303 


7.61 


2.02946 


5-94 


1.78171 


6.50 


1. 87180 


7.06 


1-95444 


7.62 


2.03078 


5-95 


1 • 78339 


6.51 


1-87334 


7.07 


1.95586 


7-63 


2.03209 


5-96 


1.78507 


6.52 


1.87487 


7.08 


I-95727 


7.64 


2.03340 


5-97 


1.78675 


6-53 


1. 87641 


7.09 


1.95869 


7-65 


2.03471 


5-98 


1 . 78842 


6-54 


1.87794 


7.10 


1 . 96009 


7.66 


2.03601 


5-99 


1 . 79009 


6-55 


1.87947 


7.11 


1. 96150 


7.67 


2.03732 


6.00 


r .79176 


6.56 


1 . 88099 


7.12 


1 .96291 


7.68 


2.03862 


6.01 


1 . 79342 


6.57 


1. 8825 1 


7-13 


1-96431 


7.69 


2.03992 


6.02 


1 . 79509 


6.58 


1 . 88403 


7.14 


1. 96571 


7.70 


2.04122 


6.03 


1 • 79675 


6.59 


1-88555 


7.15 


1 .96711 


7.71 


2.04252 


6.04 


T . 7984O 


6.60 


r 88707 


7.16 


1. 9685 1 


7.72 


2 .04381 



150 



COMPRESSED AIR 





Table 


XIII. ( 


Continued- 


-Hyperbolic Logarithms 




N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 

2. 1 1626 




rithm. 




rithm. 


7-73 


2.04511 


8.30 


8.87 


2.18267 


9.44 


2 .24496 


7-74 


2.04640 


8.31 


2. 1 1 746 


8.88 


2.18380 


9 


45 


2.24601 


7-75 


2.04769 


8.32 


2.11866 


8.89 


2.18493 


9 


46 


2.24707 


7.76 


2 . 04898 


8-33 


2. 1 1986 


8.90 


2.18605 


9 


47 


2 .24813 


7-77 


2.05027 


8-34 


2 . 12106 


8.91 


2. 18717 


9 


48 


2.24918 


7.78 


2.05156 


8-35 


2. 12226 


8.92 


2.18830 


9 


49 


2.25024 


7-79 


2.05284 


8.36 


2. 12346 


8-93 


2 . 18942 


9 


50 


2 .25129 


7.80 


2.05412 


8-37 


2.12465 


8.94 


2.19054 


9 


5i 


2.25234 


7.81 


2.05540 


8.38 


2.12585 


8-95 


2.19165 


9 


52 


2-25339 


7.82 


2.05668 


8-39 


2. 12704 


8.96 


2.19277 


9 


53 


2.25444 


7-83 


2.05796 


8.40 


2. 12823 


8-97 


2.19389 


9 


54 


2-25549 


7.84 


2.05924 


8.41 


2.12942 


8.98 


2.19500 


9 


55 


2.25654 


7.85 


2.06051 


8.42 


2. 13061 


8.99 


2. 19611 


9 


56 


2-25759 


7.86 


2 .06179 


8-43 


2. 13180 


9.00 


2. 19722 


9 


57 


2.25863 


7.87 


2.06306 


8.44 


2. 13298 


9.01 


2.19834 


9 


58 


2 .25968 


7.88 


2.06433 


8-45 


2-I34I7 


9.02 


2 . 19944 


9 


59 


2 . 26072 


7.89 


2.06560 


8.46 


2-13535 


9-03 


2.20055 


9 


60 


2 . 26176 


7.90 


2.06686 


8.47 


2-13653 


9.04 


2 .20166 


9 


61 


2 . 26280 


7.91 


2.06813 


8.48 


2-I377I 


9-o5 


2.20276 


9 


62 


2 .26384 


7.92 


2 . 06939 


8-49 


2.13889 


9.06 


2.20387 


9 


63 


2.26488 


7-93 


2.07065 


8.50 


2. 14007 


9.07 


2 .20497 


9 


64 


2 .26592 


7-94 


2.07191 


8.51 


2. 14124 


9.08 


2 . 20607 


9 


65 


2.26696 


7-95 


2.07317 


8.52 


2. 14242 


9.09 


2 . 20717 


9 


66 


2.26799 


7.96 


2.07443 


8.53 


2-14359 


9.10 


2 . 20827 


9 


67 


2 . 26903 


7-97 


2.07568 


8-54 


2. 14476 


9. 11 


2.20937 


9 


68 


2 . 27006 


7.98 


2.07694 


8-55 


2-14593 


9.12 


2 .21047 


9 


69 


2. 27109 


7-99 


2.07819 


8.56 


2. 14710 


9-13 


2.21157 


9 


70 


2 27213 


8.00 


2.07944 


8-57 


2. 14827 


9.14 


2.21266 


9 


7i 


2 . 27316 


8.01 


2 . 08069 


8.58 


2-14943 


9-i5 


2.21375 


9 


72 


2.27419 


8.02 


2 .08194 


8.59 


2. 15060 


9.16 


2.21485 


9 


73 


2 .27521 


8.03 


2.08318 


8.60 


2.15176 


9.17 


2.21594 


9 


74 


2 .27624 


8.04 


2.08443 


8.61 


2 . 15292 


9.18 


2.21703 


9 


75 


2 .27727 


8.05 


2.08567 


8.62 


2.15409 


9.19 


2 . 21812 


9 


76 


2.27829 


8.06 


2 .08691 


8.63 


2.15524 


9.20 


2 .21920 


9 


77 


2.27932 


8.07 


2.08815 


8.64 


2.15640 


9.21 


2 . 22029 


9 


78 


2 .28034 


8.08 


2.08939 


8.65 


2.15756 


9.22 


2 .22138 


9 


79 


2.28136 


8.09 


2 . 09063 


8.66 


2.15871 


9-23 


2 .22246 


9 


80 


2 .28238 


8.10 


2.09186 


8.67 


2.15987 


9.24 


2.22351 


9 


81 


2.28340 


8.11 


2.09310 


8.68 


2. 16102 


9.25 


2.22462 


9 


82 


2 .28442 


8.12 


2 • 09433 


8.69 


2. 1 62 1 7 


9.26 


2.22570 


9 


83 


2.28544 


8.13 


2.09556 


8.70 


2.16332 


9.27 


2 . 22678 


9 


84 


2.28646 


8.14 


2.09679 


8.71 


2. 16447 


9.28 


2.22786 


9 


85 


2.28747 


8.15 


2.09802 


8.72 


2 . 16562 


9.29 


2 .22894 


9 


86 


2.28849 


8.16 


2.09924 


8.73 


2. 16677 


9-30 


2 . 23001 


9 


87 


2.28950 


8.17 


2.10047 


8-74 


2.16791 


9-3i 


2.23109 


9 


88 


2.29051 


8.18 


2. 10169 


8.75 


2.16905 


9-32 


2 . 23216 


9 


89 


2.29152 


8.19 


2.10291 


8.76 


2 . 17020 


9-33 


2.23323 


9 


90 


2.29253 


8.20 


2.10413 


8.77 


2.17134 


9-34 


2.23431 


9 


91 


2.29354 


8.21 


2 -i°535 


8.78 


2. 17248 ( 


9-35 


2-23538 


9 


92 


2.29455 


8.22 


2. 10657 


8-79 


2.17361 


9-36 


2.23645 


9 


93 


2.29556 


8.23 


2.10779 


8.80 


2.17475 


9-37 


2.23751 


9 


94 


2.29657 


8.24 


2.10900 


8.81 


2.17589 


9-38 


2723858 


9 


95 


2.29757 


8.25 


2. 11021 


8.82 


2. 17702 


9-39 


2.23965 


9 


96 


2.29858 


8.26 


2. 1 1 142 


8.83 


2.17816 


9.40 


2.24071 


9 


97 


2.29958 


8.27 


2.11263 


8.84 


2 . 17929 


9.41 


2.24177 


9 


98 


2.30058 


8.28 


2. 1 1384 


8.85 


2. 18042 


9.42 


2.24284 


9 


99 


2.30158 


8.29 


2. 1 1 505 


8.86 


2.18155 


9-43 


2.24390 







TABLES 
Table XIV. — Logarithms of Numbers 



151 



No. 



IOO 
IOI 

1 02 

103 

104 

105 

106 

107 
108 

109 
no 
III 

112 

113 
114 

115 
Il6 
117 
Il8 
119 
I20 

121 
122 
123 
124 

125 
126 

127 
128 
129 

130 
131 
132 

133 

134 
135 
I36 
137 
138 

139 
I4O 
141 
142 

143 
144 

145 
I46 

147 
I48 
149 



°3 



°4 



°5 



06 



43 2 
860 

284 

7°3 
119 

53i 

938 

34: 

743 

139 

53 2 

92: 

308 

690 

070 

446 

819 



07 u 



08 



09 



555 
918 

279 
636 
991 

342 



10 037 



721 

°59 

394 
727 

°57 

385 
710 

°33 

354 
672 



13 



14 



15 



16 



i7 



301 
613 
922 
229 

534 
836 

137 
435 
732 
026 
3i9 



°43 
475 
9°3 
326 

745 
160 

57 

979 

3&3 

782 

179 

57i 

961 

346 

729 

108 

483 
856 

225 
59i 
954 

3i4 
672 

*02 6 

377 
726 

072 

4i5 
755 
°93 
428 
760 
090 
418 

743 
066 

386 

704 

*oi9 

333 

644 

953 
259 
5 6 4 
866 

167 

465 
761 

056 
348 



087 
5i8 
945 
368 
7S7 
202 
612 
*oi9 
423 
822 
218 
610 

999 

385 
767 

145 
521 
893 
262 
628 
990 

35° 
707 
*o6i 
412 
760 
106 

449 
789 
126 

461 

793 
123 

45° 
775 
098 

418 

735 
*o 5 i 

3 6 4 
675 
983 
290 
594 
897 
197 

495 
791 

085 

377 



130 

56i 

988 

410 
828 
243 

653 

*o6c 

463 
862 
258 
650 

; o 3 8 
423 
805 

183 
558 
93° 
298 
664 

*02 7 

386 
743 

^096 

447 
795 
140 

483 
823 
160 

494 
826 

156 

483 
808 
130 

45° 

767 

*o82 

395 
706 
*oi4 
320 
625 
927 
227 

524 
820 

U4 

406 



173 

604 

^030 

452 
870 
284 

694 

*IOO 

5°3 
902 

297 
689 

*o77 
461 
843 
221 

595 
967 

335 
700 
=•=063 
42 
778 

: I 3 2 
4 82 
830 
175 
517 
857 
193 
528 
860 
189 

516 

84O 
162 
481 

799 
*H4 

426 
737 
*<H5 
35i 
655 
957 
256 

554 

850 

143 
435 



217 
647 

^072 

494 
912 

3 2 5 

735 
=141 
543 
941 

33 6 

727 



! "5 

500 
881 
258 

633 

*oo4 

372 

737 

*o99 

458 

814 

*i67 

5i7 
864 
209 

55i 
890 
227 

56i 

893 
222 

548 
872 
194 

5i3 

830 

'145 

457 

768 

*o76 

381 
685 



584 
879 

173 
464 



260 
689 
: H5 
536 
953 
366 

776 
*i8i 
583 
981 
376 
766 

*I54 

538 
918 

296 
670 
041 



773 
*i35 

493 

849 



8 



55 

899 

243 

585 

924 

261 

594 
926 

254 
58i 
9°5 
226 

545 
862 
"176 
489 

799 
*io6 

412 

7i5 

017 

316 
613 
909 
202 
493 



3°3 
732 
: i57 
578 
995 
407 

816 

=222 

623 

*02I 

415 
805 

^192 

576 

95 6 

333 

707 
#078 

445 

♦171 

529 

884 

*237 

587 
934 
278 

619 
958 
294 

628 

959 
28 

613 

937 
258 

577 

893 

*2o8 

520 
829 

*i37 
442 
746 

*047 

346 

643 
938 
231 
522 



346 
775 
199 

620 

$036 
449 
857 

*2 62 
663 

*o6o 
454 
844 
231 
614 
994 

37i 
744 
ii5 
482 
846 

*2C>7 

565 

920 

*272 
621 
968 
312 

653 
992 
327 
66l 
992 
32O 
646 
969 
29O 
609 
925 
"239 

551 



817 

*242 

662 

+078 
490 
898 
=302 

7°3 

*IOO 

493 

883 

*269 
652 

*Q32 



♦151 

5i8 
882 

*243 
600 
955 

*3°7 
656 

*oo3 
346 
687 

*025 

361 
694 

*024 

352 
678 

*OOI 

322 

640 

95 6 

^270 



*i68 

473 
776 

*o77 

376 

673 
967 

260 

55i 



Pp. Pts. 



5°3 
806 
*io7 
406 
702 
997 
289 
580 



41 

4-1 
8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32-8 
36.9 



38 

3-8 
7.6 
11. 4 

15-2 

19.0 
22.8 
26.6 
3°-4 
34-2 



42 
4-2 

8.4 
12.6 
16.8 
21.0 

25-2 
29.4 

33-6 
38.7 37.8 



40 

4.0 
8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
3?.o 
36.0 



37 
3-7 
7-4 
11. 1 
14.8 
18. 5 
22.2 
25-9 
29.6 
33-3 



35 


34 


3-5 
7.0 


3 
6 


4 
8 


10.5 


10 


2 


14.0 


13 


6 


17-5 


17 





21.0 


20 


4 


24-5 
28.0 


23 
27 


<S 
2 


31-5 


3° 


6 



36 

3-6 

7.2 
10.8 
14.4 
18.0 

21.6 
25.2 
28.8 
32.4 



33 

3-3 
6.6 

9-9 
13.2 
16. s 
19.8 
23.1 
26.4 
29.7 



32 
3-: 
6.. 
9.6 
12. 
16.0 
19.2 
22.4 
25.6 
28.8 



31 

3-1 
6.2 
9-3 

12.4 
iS-5 
18.6 
21.7 
24.8 
27.9 



152 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


ISO 


17 609 


638 


667 


696 


725 


754 


782 


811 


840 


869 




151 


898 


926 


955 


984 


*oi3 


*c>4i 


*o7o 


*o99 


*I27 


*i 5 6 




152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


441 




29 


28 


153 


469 


498 


526 


554 


583 


611 


6 39 


667 


696 


724 


1 
2 


2.9 

5 8 


2.8 

5-6 
8.4 


154 


752 


780 


808 


837 


865 


893 


921 


949 


977 


*oc>5 


3 


8.7 


iS5 


19 o33 


061 


089 


117 


145 


173 


201 


229 


257 


285 


4 


11. 6 


11. 2 


156 


312 


340 


368 


396 


424 


451 


479 


5°7 


535 


562 


5 
6 


14-5 
17.4 


16^8 


157 


59o 


618 


645 


673 


700 


728 


756 


783 


8n 


838 


7 


20.3 


19.6 


158 


866 


893 


921 


948 


976 


*oo3 


*o3o 


*o58 


*o85 


*II2 


8 
9 


23.2 
26. 1 


22.4 
25. 2 


159 


20 140 


167 


194 


222 


249 


276 


3°3 


33° 


358 


385 




160 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




161 


683 


710 


737 


7 6 3 


790 


817 


844 


871 


898 


925 




162 


95 2 


978 


*oo5 


*C>32 


*°59 


*o8 S 


*H2 


*i39 


*i6 5 


*I92 


1 


27 

2.7 
5-4 


26 

2.6 


163 


21 219 


245 


272 


299 


325 


352 


378 


405 


43i 


458 


2 


5-2 


164 


484 


5" 


537 


564 


59° 


617 


6 43 


669 


696 


722 


3 


8.1 


7.8 


165 


748 


775 


801 


827 


854 


880 


906 


932 


958 


985 


4 

5 


10.8 
13-5 


10.4 

13-0 


166 


22 on 


°37 


063 


089 


ii5 


141 


167 


194 


220 


246 


6 


16.2 


15.6 


167 


272 


298 


324 


35° 


376 


401 


427 


453 


479 


505 


7 
8 


18.9 
21.6 


18.2 
20.8 


168 


531 


557 


583 


608 


634 


660 


686 


712 


737 


7 6 3 


9 


24-3 


23.4 


169 


789 


814 


840 


866 


891 


917 


943 


963 


994 


*oi9 




170 


23 °45 


070 


096 


121 


147 


172 


198 


223 


249 


274 




171 


300 


325 


35° 


37 6 


401 


426 


452 


477 


5° 2 


528 


id 


172 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


I 


2.5 


173 


805 


830 


855 


880 


9°5 


93° 


955 


980 


*oo5 


^030 


2 


s 





174 


24 o55 


080 


105 


130 


i55 


180 


204 


229 


254 


279 


3 
4 


7 

10 


5 



175 


3°4 


329 


353 


378 


403 


428 


. 452 


477 


502 


527 


5 


12 


S 


176 


55i 


576 


601 


625 


650 


674 


699 


724 


748 


773 


6 

7 


IS 
17 




5 


177 


797 


822 


846 


871 


895 


920 


944 


969 


993 


*oi8 


8 


20 





178 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


9 


22.5 


179 


285 


310 


334 


358 


382 


406 


43i 


455 


479 


5°3 




180 


527 


55i 


575 


600 


624 


648 


672 


696 


720 


744 




181 


768 


792 


816 


840 


864 


888 


912 


935 


959 


983 




24 


23 


182 


26 007 


031 


°55 


079 


102 


126 


i5° 


i74 


198 


221 


1 


2.4 


2.3 


183 


245 


269 


293 


316 


34o 


3 6 4 


387 


411 


435 


458 


2 
3 


4.8 
7. 2 


4.6 
6.9 


184 


482 


505 


529 


553 


576 


600 


623 


647 


670 


694 


4 


9.6 


9.2 


185 


717 


74i 


764 


788 


811 


834 


858 


881 


905 


928 


5 
6 

7 


12.0 


11. 5 

T? 9. 


186 


95i 


975 


998 


*02I 


*°45 


*o68 


*09i 


*H4 


* I3 8 


*i6i 


14.4 
16.8 


13.0 

16. 1 


187 


27 184 


207 


231 


254 


277 


300 


323 


346 


37° 


393 


8 


19. 2 


18.4 


188 


416 


439 


462 


485 


5°8 


53i 


554 


577 


600 


623 


9 21.0 20. 7 


189 


646 


669 


692 


715 


738 


761 


784 


807 


830 


852 




190 


875 


898 


921 


944 


967 


989 


*OI2 


*°35 


*o58 


*o8i 




191 


28 103 


126 


149 


171 


194 


' 217 


24O 


262 


285 


3°7 




22 


21 


192 


33° 


353 


375 


398 


421 


443 


466 


488 


5" 


533 


1 


2. 2 

4-4 
6.6 


2. I 


193 


556 


578 


601 


623 


646 


668 


691 


7i3 


735 


758 


3 


i'-l 


194 


780 


803 


825 


847 


870 


892 


914 


937 


959 


981 


4 
5 
6 


8.8 


8.4 


195 


29 003 


026 


048 


070 


092 


ii5 


137 


159 


181 


203 


13-2 


10. 5 
12.6 


196 


226 


248 


270 


292 


3i4 


33 6 


358 


380 


403 


425 


7 


iS-4 


14.7 


197 


447 


469 


491 


513 


535 


557 


579 


601 


623 


645 


8 
9 


17.6 
10.8 


16.8 
18.0 


198 


667 


688 


710 


732 


754 


776 


798 


820 


842 


863 




199 


885 


907 


929 


95i 


973 


994 


*oi6 


*o38 


*o6o 


*o8i 






30 























TABLES 

Table XIV. Continued. — Logarithms of Numbers 



153 



No. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


200 


30 103 


125 


146 


168 


190 


211 


233 


255 


276 


298 




20 1 


320 


34i 


3 6 3 


384 


406 


428 


449 


471 


492 


5i4 




202 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 


22 


21 


203 


75° 


771 


792 


814 


835 


856 


878 


899 


920 


942 


1 2. 

2 4. 

3 6. 


2 2.1 

* 4-2 
5 6.3 


204 


963 


984 


*oo6 


*02 7 


*ch8 


*o69 


*o9i 


*II2 


*i33 


*i54 


205 


31 175 


197 


218 


239 


260 


281 


302 


323 


345 


366 


4 8. 


i 8.4 


206 


387 


408 


429 


45° 


471 


492 


513 


534 


555 


576 


5 ii- 

6 13. 


3 10. s 
212.6 


207 


597 


618 


639 


660 


681 


702 


7 2 3 


744 


7 6 5 


785 


7 is.- 


4 14-7 


208 


806 


827 


848 


869 


890 


911 


93i 


952 


973 


994 


8 17. 

9 19- 


6 16. S 
S 18.0 


209 


3 2 OI 5 


°35 


056 


077 


098 


118 


139 


160 


181 


201 




2IO 


222 


243 


263 


284 


3°5 


325 


346 


366 


387 


408 




211 


428 


449 


469 


490 


5io 


53i 


55'2 


572 


593 


613 




212 


634 


654 


675 


695 


7i5 


736 


756 


777 


797 


818 




20 


213 


838 


858 


879 


899 


919 


940 


960 


980 


*OOI 


*02I 


2 


4.0 


214 


33 041 


062 


082 


102 


122 


143 


163 


183 


203 


224 


3 


6.0 


215 


244 


264 


284 


3°4 


325 


345 


365 


385 


405 


425 


i 


8. 
10. 


2l6 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 


6 


12.0 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


826 


7 
g 


14.0 
16.0 


2l8 


846 


866 


885 


9°5 


925 


945 


965 


985 


*oo5 


*025 


9 


18.0 


219 


34 044 


064 


084 


104 


124 


143 


163 


183 


203 


223 




220 


242 


262 


282 


301 


321 


34i 


361 


380 


400 


420 




221 


439 


459 


479 


498 


5i8 


537 


557 


577 


596 


616 




19 
1.9 


222 


635 


655 


674 


694 


7i3 


733 


753 


772 


792 


811 


1 


223 


830 


850 


869 


889 


908 


928 


947 


967 


986 


*oo5 


2 


3-8 


224 


35 ° 2 5 


044 


064 


083 


102 


122 


141 


160 


180 


199 


3 

4 


5-7 
7.6 


225 


218 


238 


257 


276 


295 


3i5 


334 


353 


372 


392 


5 


9-5 


226 


411 


43° 


449 


468 


488 


5°7 


526 


545 


5 6 4 


583 


6 

7 


11. 4 
13.3 


227 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 


8 


15-2 


228 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 


9 


17. 1 


229 


984 


*oo3 


*02I 


*040 


*°59 


=1=078 


*o97 


*n6 


*i35 


*i54 




23O 


36 173 


192 


211 


229 


248 


267 


286 


3°5 


324 


342 




231 


361 


380 


399 


418 


43 6 


455 


474 


493 


5ii 


53° 




18 


232 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


1 


1.8 


233 


736 


754 


773 


791 


810 


829 


847 


866 


884 


9°3 


2 
3 


3-6 
5-4 


234 


922 


940 


959 


977 


996 


*oi4 


*o 33 


*°5i 


*o"jo 


*o88 


4 


7-2 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


5 
6 


9.0 

10.8 


236 


291 


310 


328 


346 


3 6 5 


383 


401 


420 


438 


457 


7 


12.6 


237 


475 


493 


5ii 


53° 


548 


566 


585 


603 


621 


639 


8 


14.4 
16. 2 


238 


658 


676 


694 


712 


73i 


749 


767 


785 


803 


822 


9 


239 


840 


858 


876 


894 


912 


93i 


949 


967 


985 


*oo3 




24O 


38 021 


°39 


o57 


o75 


°93 


112 


130 


148 


166 


184 




241 


202 


220 


238 


256 


2 74 


292 


310 


328 


346 


3 6 4 




17 


242 


382 


399 


417 


435 


453 


47i 


489 


5°7 


525 


543 


1 
2 


i-7 

3.4 


243 


56i 


578 


596 


614 


632 


650 


668 


686 


7°3 


721 


3 


5-i 


244 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


4 
5 
6 


6.8 
8-5 
10. 2 


245 


917 


934 


952 


970 


987 


*oo5 


*023 


*04i 


*o58 


*c>76 


246 


39 °94 


in 


129 


146 


164 


182 


199 


217 


235 


252 


7 
8 
9 


11. 9 
13.6 
15-3 


247 


270 


287 


305 


322 


34o 


358 


375 


393 


410 


428 


248 


445 


463 


480 


498 


5i5 


533 


55° 


568 


585 


602 




249 


620 


6 37 


655 


672 


690 


707 


724 


742 


759 


777 





154 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



"No. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


250 


39 794 


811 


829 


846 


863 


881 


898 


9i5 


933 


95° 




251 


967 


985 


*002 


*oi9 


*o 3 7 


*o54 


*o7i 


*o88 


*io6 


*I2 3 




252 


40 140 


157 


175 


192 


209 


226 


243 


261 


278 


295 




18 


253 


312 


329 


346 


3 6 4 


381 


398 


4i5 


432 


449 


466 


I 


1.8 
3-6 

5-4 


254 


483 


500 


5l8 


535 


552 


5 6 9 


586 


603 


620 


637 


3 


255 


654 


671 


688 


7°5 


722 


739 


756 


773 


790 


807 


4 


7.2 


256 


824 


841 


858 


875 


892 


909 


926 


943 


960 


976 


5 
6 


9.0 
10.8 


257 


993 


*OIO 


*02 7 


*o44 


*o6i 


#078 


*o 9 5 


*iii 


*I28 


*I45 


7 


12.6 


258 


41 162 


179 


196 


212 


229 


246 


263 


280 


296 


3*3 


8 

Q 


14.4 


259 


33° 


347 


3 6 3 


380 


397 


414 


43° 


447 


464 


481 




260 


497 


5i4 


53i 


547 


5 6 4 


581 


597 


614 


631 


647 




261 


664 


681 


697 


714 


73i 


747 


764 


780 


797 


814 




262 


830 


847 


863 


880 


896 


913 


929 


946 


963 


979 




17 


263 


996 


*OI2 


*029 


*Q45 


*o62 


=1=078 


*°95 


*iii 


*I27 


*I44 


2 


1-7 

3-4 


264 


42 160 


177 


193 


210 


226 


243 


259 


275 


292 


308 


3 


5-1 


265 


3 2 5 


341 


357 


374 


39° 


406 


423 


439 


455 


472 


4 

5 


6.8 
8-5_ 


266 


488 


5°4 


521 


. 537 


553 


57° 


586 


602 


619 


635 


6 


10. 2 


267 


6Si 


667 


684 


700 


716 


732 


749 


765 


781 


797 


7 
8 


11. 9 


268 


813 


830 


846 


862 


878 


894 


911 


927 


943 


959 


9 


IS- 3 


269 


975 


991 


*ooS 


*024 


*04o 


*o56 


+072 


*o88 


*io4 


*I20 




270 


43 i3 6 


152 


169 


185 


201 


217 


233 


249 


265 


28l 




271 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 




16 
1.6 


272 


457 


473 


489 


5°5 


52i 


537 


553 


5 6 9 


584 


60O 


1 


273 


616 


632 


648 


664 


680 


696 


712 


727 


743 


759 


2 


3 


2 


274 


775 


791 


807 


823 


838 


854 


870 


886 


902 


917 


3 

4 


4 
6 


8 
4 


275 


933 


949 


965 


981 


996 


*OI2 


*028 


*o44 


*°59 


*°75 


5 


8 


Q 


276 


44 091 


107 


122 


138 


154 


170 


185 


201 


217 


232 


6 

7 

8 


9 


6 


277 


248 


264 


2 79 


295 


3" 


326 


342 


358 


373 


389 


12 


8 


278 


404 


420 


43 6 


45i 


467 


483 


498 


514 


529 


545 


9 


14 


4 


279 


560 


576 


592 


607 


623 


638 


654 


669 


685 


700 




280 


716 


73i 


747 


762 


778 


793 


809 


824 


840 


855 




281 


871 


886 


902 


917 


932 


948 


963 


979 


994 


*OIO 




15 


282 


45 ° 2 5 


040 


056 


071 


086 


102 


117 


t-33 


148 


163 


1 


1-5 


283 


179 


194 


209 


225 


240 


255 


271 


286 


301 


317 


2 
3 


3 
4 



5 


284 


332 


347 


362 


378 


393 


408 


423 


439 


454 


469 


4 


6 





285 


484 


500 


5i5 


53° 


545 


56i 


576 


59i 


606 


621 


5 
6 

7 


7 


S 


286 


637 


652 


667 


682 


697 


712 


728 


743 


758 


773 


9 

10 


s 


287 


788 


803 


818 


834 


849 


864 


879 


894 


909 


924 


8 


12 





288 


939 


954 


969 


984 


*ooo 


*oi5 


*o3o 


*<H5 


*o6o 


*o75 


9 


13 5 


289 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 




290 


240 


255 


270 


285 


300 


3i5 


33° 


345 


359 


374 




291 


389 


404 


419 


434 


449 


464 


479 


494 


5°9 


523 




14 


292 


538 


553 


568 


583 


598 


613 


627 


642 


657 


672 


1 
2 


1 


4 
8 


293 


687 


702 


716 


73i 


746 


761 


776 


790 


805 


820 


3 


4 


2 


294 


835 


850 


864 


879 


894 


909 


923 


938 


953 


967 


4 


5 


6 


295 


982 


997 


*OI2 


*02 6 


*04i 


=•=056 


+070 


*o85 


*IOO 


*U4 


5 
6 


7 
8 


4 


296 


47 I2 9 


144 


159 


173 


188 


202 


217 


232 


246 


261 


7 
8 
9 


9 


8 


297 


276 


290 


3°5 


319 


334 


349 


3 6 3 


378 


392 


407 


11 

12 


2 
6 


298 


422 


43 6 


45i 


465 


480 


494 


5°9 


524 


538 


553 




299 


567 


582 


596 


611 


625 


640 


654 


669 


683 


698 





TABLES 
Table XIV. Continued. — Logarithms of Numbers 



155 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


3OO 


47 7 12 


727 


74i 


75 6 


770 


784 


799 


813 


828 


842 




301 


857 


871 


885 


900 


914 


929 


943 


958 


972 


986 




302 


48 001 


015 


029 


044 


058 


°73 


087 


IOI 


116 


130 




303 


144 


i59 


173 


187 


202 


216 


230 


244 


259 


2 73 




304 


287 


302 


316 


33° 


344 


359 


373 


387 


401 


416 




305 


43° 


444 


458 


473 


487 


5 DI 


5i5 


53° 


544 


558 




IS 


306 


572 


586 


601 


615 


629 


643 


657 


671 


686 


700 


2 


3 





307 


714 


728 


742 


756 


770 


785 


799 


813 


827 


841 


3 


4 


5 


308 


855 


869 


883 


897 


911 


926 


940 


954 


968 


982 


4 
5 


6 

7 



5 


309 


996 


*OIO 


*024 


*o 3 8 


*052 


*o66 


*o8o 


*o94 


*io8 


*I22 


6 


9 





3IO 


49 *3 6 


150 


164 


178 


192 


206 


220 


234 


248 


262 


7 

8 


10 


5 


311 


276 


290 


3°4 


3i8 


332 


346 


360 


374 


388 


402 


9 


13 


5 


312 


415 


429 


443 


457 


471 


485 


499 


5i3 


5 2 7 


541 




313 


554 


568 


582 


596 


610 


624 


638 


651 


665 


679 




314 


693 


707 


721 


734 


748 


762 


776 


790 


803 


817 




315 


831 


845 


859 


872 


S86 


900 


914 


927 


941 


955 




316 


969 


982 


996 


*OIO 


*024 


*°37 


*o5i 


*o65 


*o79 


*CX)2 




14 


317 


50 106 


120 


133 


147 


161 


i74 


188 


262 


215 


229 


1 
2 


1 
2 


4 
8 


318 


243 


256 


270 


284 


297 


3" 


325 


338 


352 


3 6 5 


3 


4 


2 


319 


379 


393 


406 


420 


433 


447 


461 


474 


4S8 


5°i 


4 
5 
6 


5 


6 


320 


5i5 


529 


542 


556 


5 6 9 


583 


59 6 


610 


623 


637 


7 
8 


4 


321 


651 


664 


678 


691 


7°5 


718 


732 


745 


759 


772 


7 

g 


9 


8 


322 


786 


799 


813 


826 


840 


853 


866 


880 


893 


907 


9 


12 


6 


323 


920 


934 


947 


961 


974 


987 


*OOI 


*oi4 


*028 


*o4i 




324 


5i °55 


068 


081 


095 


108 


121 


135 


148 


162 


175 




325 


188 


202 


215 


228 


242 


255 


268 


282 


295 


308 




326 


322 


335 


348 


362 


375 


388 


402 


415 


428 


441 




327 


455 


468 


481 


495 


508 


52i 


534 


548 


561 


574 




13 


328 


587 


601 


614 


627 


640 


654 


667 


680 


693 


706 


1 


1 


3 
6 


329 


720 


733 


746 


759 


772 


786 


799 


812 


825 


838 


3 


3 


9 


330 


851 


865 


878 


891 


904 


917 


93° 


943 


957 


970 


4 


5 


2 


331 


983 


996 


*oo9 


*022 


*°35 


*o4S 


*o6i 


*o75 


*o88 


*IOI 


5 
6 


7 


5 
8 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


231 


7 
8 
9 


9 


1 


333 


244 


257 


270 


284 


297 


310 


323 


336 


349 


362 


10 
11 


4 
7 


334 


375 


388 


401 


414 


427 


440 


453 


466 


479 


492 




335 


5°4 


5i7 


53° 


543 


556 


569 


582 


595 


608 


621 




336 


634 


647 


660 


673 


686 


699 


711 


724 


737 


75° 




337 


763 


776 


789 


802 


815 


827 


840 


853 


866 


879 




338 


892 


9°5 


917 


93° 


943 


95 6 


969 


982 


994 


*oo7 




12 


339 


53 020 


°33 


046 


058 


071 


084 


097 


no 


122 


135 


1 


I 


2 


340 


148 


161 


i73 


186 


199 


212 


224 


237 


250 


263 


3 


3 


6 


34i 


275 


288 


301 


3i4 


326 


339 


352 


3 6 4 


377 


39° 


4 


4 
6 

7 


8 


342 


403 


4i5 


428 


441 


453 


466 


479 


491 


5°4 


5i7 


5 

6 


2 


343 


529 


542 


555 


567 


580 


593 


605 


618 


631 


643 


7 


8 


4 


344 


656 


668 


681 


694 


706 


719 


732 


744 


757 


769 


8 

Q 


9 

TCJ 


6 
8 


345 


782 


794 


807 


820 


832 


845 


857 


870 


882 


895 




346 


908 


920 


933 


945 


958 


970 


983 


995 


*oo8 


*020 




347 


54 033 


o45 


058 


070 


083 


, 095 


108 


120 


133 


145 




348 


158 


170 


183 


195 


208 


220 


233 


245 


258 


270 




349 


283 


295 


3°7 


320 


332 


345 


357 


37° 


382 


394 





156 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


350 


54 407 


419 


43 2 


444 


456 


469 


481 


494 


506 


5i8 




351 


53i 


543 


555 


568 


580 


593 


605 


617 


630 


642 




352 


654 


667 


679 


691 


704 


716 


728 


741 


753 


765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 


888 




354 


900 


913 


925 


937 


949 


962 


974 


986 


998 


*OII 




355 


55 ° 2 3 


°35 


047 


060 


072 


084 


096 


108 


121 


T-33 




13 


356 


145 


157 


169 


182 


194 


206 


218 


230 


242 


255 


1 

2 


2 


A 

6 


357 


267 


279 


291 


3°3 


3i5 


328 


34o 


352 


3 6 4 


376 


3 


3 


9 


358 


388 


400 


413 


425 


437 


449 


461 


473 


485 


497 


4 
5 


5 
6 


2 

s 


359 


5°9 


522 


534 


546 


558 


57° 


582 


594 


606 


618 


6 


7 


8 


360 


630 


642 


654 


666 


678 


691 


7°3 


7i5 


727 


739 


7 
8 


9 


1 


361 


75i 


763 


775 


787 


799 


811 


823 


835 


847 


859 


9 


II 


4 
7 


362 


871 


883 


895 


907 


919 


93i 


943 


955 


967 


979 




363 


991 


*oo3 


*oi5 


*02 7 


*o 3 8 


*o5o 


*o62 


*o74 


*o86 


*098 




364 


56 no 


122 


134 


146 


158 


170 


182 


194 


205 


217 




365 


229 


241 


253 


265 


277 


289 


301 


312 


324 


33 6 




366 


348 


360 


372 


384 


396 


407 


419 


43i 


443 


455 




12 


367 


467 


478 


490 


502 


514 


526 


538 


549 


56i 


573 


1 
2 


I 
2 


2 
4 


368 


585 


597 


608 


620 


632 


644 


656 


667 


679 


691 


3 


3 


6 


369 


7°3 


7i4 


726 


738 


75° 


761 


773 


785 


797 


808 


4 
5 
6 


4 
6 


8 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


7 


2 


371 


937 


949 


961 


972 


984 


996 


*oo8 


*oi9 


*o3i 


*043 


7 
g 


8 


4 
6 


372 


57 °54 


066 


078 


089 


IOI 


"3 


124 


136 


148 


159 


9 


9 

10 


8 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 


276 




374 


287 


299 


310 


322 


334 


345 


357 


368 


380 


392 




375 


403 


415 


426 


438 


449 


461 


473 


484 


496 


5°7 




376 


5i9 


53° 


542 


553 


565 


576 


588 


600 


611 


623 




377 


634 


646 


6 57 


669 


680 


692 


7°3 


7i5 


726 


738 




II 


378 


749 


761 


772 


784 


795 


807 


818 


830 


841 


852 


1 


I 


1 


379 


864 


875 


887 


898 


910 


921 


933 


944 


955 


967 


3 


3 


3 


380 


978 


990 


*OOI 


*oi3 


*024 


*o 3 5 


*o47 


*o58 


*o7o 


*o8i 


4 
5 
6 


4 


4 


38i 


58 092 


104 


ii5 


127 


138 


149 


161 


172 


184 


195 


5 
6 


5 
6 


382 


206 


218 


229 


240 


252 


263 


274 


286 


297 


3°9 


7 
8 
9 


7 


7 


383 


320 


33i 


343 


354 


365 


377 


388 


399 


410 


422 


8 



8 



384 


433 


444 


45 6 


467 


478 


490 


5°i 


512 


524 


535 




385 


546 


557 


5 6 9 


580 


591 


602 


614 


625 


636 


647 




386 


659 


670 


681 


692 


704 


7i5 


726 


737 


749 


760 




387 


771 


782 


794 


805 


816 


827 


838 


850 


861 


872 




388 


883 


894 


906 


917 


928 


939 


95° 


961 


973 


984 




10 


389 


995 


*oo6 


*oi7 


*028 


*040 


*°5i 


*o62 


*o73 


*o84 


*°95 


1 


I 





390 


59 i°6 


118 


129 


140 


151 


162 


173 


184 


195 


207 


3 


3 





39i 


218 


229 


240 


251 


262 


273 


284 


295 


306 


3i8 


4 


4 





392 


3 2 9 


340 


35i 


362 


373 


384 


395 


406 


4i7 


428 


5 
6 


S 
6 






393 


439 


45° 


461 


472 


483 


494 


506 


5i7 


528 


539 


7 


7 





394 


55° 


56i 


572 


583 


594 


605 


616 


627 


638 


649 


8 
g 


8 

n 






395 


660 


671 


682 


6 93 


704 


7i5 


726 


737 


748 


759 




396 


770 


780 


791 


802 


8i3 


824 


835 


846 


857 


868 




397 


879 


890 


901 


912 


923 


934 


945 


95 6 


966 


977 




398 


988 


999 


*OIO 


*02I 


*032 


*043 


*o54 


*o65 


+076 


*o86 




399 


60 097 


108 


119 


I30 


141 


152 


163 


173 


184 


i95 





TABLES 

Table XIV. Continued. — Logarithms of Numbers 



157 



No. 





1 


2 


3 


4 


5 


6 7 


8 


9 


Pp. Pts. 


400 


60 206 


217 


228 


239 


249 


260 


271 


282 


293 


3°4 




401 


3i4 


325 


33 6 


347 


358 


3 6 9 


379 


39° 


401 


412 




402 


423 


433 


444 


455 


466 


477 


487 


498 


5°9 


520 




403 


53i 


54i 


552 


5 6 3 


574 


584 


595 


606 


617 


627 




404 


638 


649 


660 


670 


681 


692 


7°3 


7i3 


724 


735 




405 


746 


756 


767 


778 


788 


799 


810 


821 


831 


842 




406 


853 


863 


874 


885 


895 


906 


917 


927 


938 


949 




407 


959 


970 


981 


991 


*002 


*oi3 


*023 


*°34 


*°45 


*o55 




408 


61 066 


077 


087 


098 


IO9 


119 


130 


140 


151 


162 


I 


11 

1.1 


409 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 


2 


2.2 


41O 


278 


289 


300 


310 


321 


33* 


342 


352 


3 6 3 


374 


3 

4 
5 


3-3 

4-4 
5-5 


4 II 


384 


395 


405 


416 


426 


437 


448 


458 


469 


479 


412 


490 


500 


5" 


521 


532 


542 


553 


5 6 3 


574 


584 


6 
7 
8 


6.6 
7-7 
8.8 


413 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 


414 


700 


711 


721 


73i 


742 


752 


7 6 3 


773 


784 


794 


9 


9.9 


415 


805 


815 


826 


836 


847 


857 


868 


87S 


888 


899 




416 


909 


920 


93° 


941 


951 


962 


972 


982 


993 


*oc>3 




417 


62 014 


024 


034 


°45 


°55 


066 


076 


0S6 


097 


107 




418 


118 


128 


138 


149 


i59 


170 


180 


190 


201 


211 




419 


221 


232 


242 


252 


263 


273 


284 


294 


3°4 


315 




420 


3 2 5 


335 


346 


356 


366 


377 


387 


397 


408 


418 




421 


428 


439 


449 


459 


469 


480 


490 


500 


5" 


521 




10 


422 


53i 


542 


552 


562 


572 


583 


593 


603 


613 


624 


I 


1.0 


423 


634 


644 


655 


665 


675 


685 


696 


706 


716 


726 


2 


2.0 


424 


737 


747 


757 


767 


778 


788 


798 


808 


818 


829 


3 

4 


3-° 
4.0 


425 


839 


849 


859 


870 


880 


890 


900 


910 


921 


93i 


5 


5-o 


426 


941 


951 


961 


972 


982 


992 


*002 


*OI2 


*022 


*o33 


6 

7 


6.0 
7.0 


427 


63 043 


•053 


063 


o73 


083 


094 


IO4 


114 


124 


134 


8 


8.0 


428 


144 


155 


165 


175 


185 


i95 


205 


215 


225 


236 


9 


9.0 


429 


246 


256 


266 


276 


286 


296 


306 


317 


327 


337 




430 


347 


357 


3 6 7 


377 


387 


397 


407 


417 


428 


438 




431 


448 


458 


468 


478 


488 


498 


508 


518 


528 


538 




432 


548 


558 


568 


579 


589 


599 


699 


619 


629 


639 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 




435 


849 


859 


869 


879 


889 


899 


909 


919 


929 


939 




9 


436 


949 


959 


969 


979 


988 


998 


*oo8 


*oi8 


*028 


#038 


1 


0.9 


437 


64 048 


058 


068 


078 


088 


098 


108 


118 


128 


137 


2 


1.8 


438 


147 


157 


167 


177 


187 


197 


207 


217 


227 


237 


3 

4 


2 . 7 
3-6 


439 


246 


256 


266 


276 


286 


296 


306 


316 


326 


335 


5 
g 


4-5 
5-4 
6.3 


440 


345 


355 


365 


375 


385 


395 


404 


414 


424 


434 


7 


441 


444 


454 


464 


473 


483 


493 


5°3 


5i3 


523 


532 


8 


7.2 
8.1 


442 


542 


552 


562 


572 


582 


59i 


601 


611 


621 


631 


9 


443 


640 


650 


660 


670 


680 


689 


699 


709 


719 


729 




444 


738 


748 


758 


768 


777 


787 


797 


807 


816 


826 




445 


836 


846 


856 


865 


875 


885 


895 


904 


914 


924 




446 


933 


943 


953 


9 6 3 


972 


982 


992 


*002 


*OII 


*02I 




447 


65 031 


040 


050 


060 


070 


079 


089 


O99 


108 


Il8 




448 


128 


137 


147 


157 


167 


176 


186 


I96 


205 


21S 




449 


225 


234 


244 


254 


263 


2 73 


283 


292 


302 


312 





158 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


450 


65 321 


33i 


34i 


35° 


360 


3 6 9 


379 


389 


398 


408 




451 


418 


427 


437 


447 


45 6 


466 


475 


485 


495 


5°4 




452 


514 


523 


533 


543 


552 


562 


57i 


581 


59i 


600 




453 


610 


619 


629 


639 


648 


658 


667 


677 


686 


696 




454 


706 


7i5 


725 


734 


744 


753 


763 


772 


782 


792 




455 


801 


811 


820 


830 


839 


849 


858 


868 


877 


887 




456 


896 


906 


916 


925 


935 


944 


954 


963 


973 


982 




457 


992 


*OOI 


*OII 


*020 


#030 


!=o 39 


*049 


=1=058 


*o68 


*o77 




458 


66 087 


096 


106 


115 


124 


134 


143 


153 


162 


172 


1 


10 

1 . 


459 


181 


191 


200 


2IO 


219 


229 


238 


247 


257 


266 


2 


2.0 


460 


276 


285 


295 


3°4 


3i4 


323 


332 


342 


35i 


361 


3 


3-° 


461 


37° 


380 


389 


398 


408 


417 


427 


43 6 


445 


455 


4 
5 


4.0 
5-0 


462 


464 


474 


483 


492 


502 


5" 


52i 


53° 


539 


549 


6 


6.0 


463 


558 


5 6 7 


577 


586 


596 


605 


614 


624 


633 


642 


7 
8 


7.0 
8.0 


464 


652 


661 


671 


680 


689 


699 


708 


717 


727 


736 


9 


9.0 


465 


745 


755 


764 


773 


783 


792 


801 


811 


820 


829 




466 


839 


848 


857 


867 


876 


885 


894 


904 


9*3 


922 




467 


93 2 


941 


95° 


960 


969 


978 


987 


997 


*oo6 


*°i5 




468 


67 025 


034 


043 


052 


062 


071 


080 


089 


099 


108 




469 


117 


127 


136 


145 


154 


164 


173 


182 


191 


201 




470 


210 


219 


228 


237 


247 


256 


265 


274 


284 


293 




47i 


302 


311 


321 


33° 


339 


348 


357 


367 


376 


385 




472 


394 


403 


413 


422 


43i 


440 


449 


459 


468 


477 


1 


0.9 


473 


486 


495 


5°4 


5i4 


523 


532 


54i 


55° 


560 


5 6 9 


2 


1.8 


474 


578 


587 


596 


605 


614 


624 


633 


642 


651 


660 


3 

4 


2.7 
3-6 


475 


669 


679 


688 


697 


706 


7i5 


724 


733 


742 


752 


5 


4-5 


476 


761 


770 


779 


788 


797 


806 


815 


825 


834 


843 


6 

7 
8 


5-4 
6.3 


477 


852 


861 


870 


879 


888 


897 


906 


916 


925 


934 


7.2 


478 


943 


95 2 


961 


970 


979 


988 


997 


*oo6 


*oi5 


*024 


9 


8.1 


479 


68 034 


043 


052 


061 


070 


079 


088 


097 


106 


115 




480 


124 


i33 


142 


151 


160 


169 


178 


187 


196 


205 




481 


215 


224 


233 


242 


251 


260 


269 


278 


287 


296 




482 


3°5 


3i4 


323 


332 


34i 


35° 


359 


368 


377 


386 




483 


395 


404 


413 


422 


43i 


440 


449 


458 


467 


476 




484 


485 


494 


502 


5ii 


520 


529 


538 


547 


556 


56s 




485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


655 




g 


486 


664 


673 


681 


690 


699 


708 


717 


726 


735 


744 


I 


0.8 


487 


753 


762 


771 


780 


789 


797 


806 


8i5 


824 


833 


2 


1.6 


488 


842 


851 


860 


869 


878 


886 


895 


904 


9i3 


922 


3 

4 


2.4 
3-2 


489 


93i 


940 


949 


958 


966 


975 


984 


993 


*002 


*OII 


5 
6 

7 


4.0 
4.8 
5-6 


490 


69 020 


028 


°37 


046 


°55 


064 


°73 


082 


090 


099 


491 


108 


117 


126 


135 


144 


i5 2 


161 


170 


179 


188 


8 


6.4 


492 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 


9 


7.2 


493 


285 


294 


302 


3ii 


320 


329 


338 


346 


355 


3 6 4 




494 


373 


381 


39° 


399 


408 


4i7 


425 


434 


443 


45 2 . 




495 


461 


469 


478 


487 


496 


5°4 


513 


522 


53i 


539 




496 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 




497 


636 


644 


653 


662 


671 


679 


688 


697 


7°5 


7i4 




498 


723 


732 


740 


749 


758 


767 


775 


784 


793 


801 




499 


810 


819 


827 


836 


845 


854 


862 


871 


880 


888 





TABLES 

Table XIV. Continued. — Logarithms of Numbers 



159 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pt». 


500 


69 897 


906 


914 


923 


93 2 


940 


949 


958 


966 


975 




50I 


984 


992 


<=OOI 


l=OIO : 


1=018 " 


1=027 : 


^03 6 " 


1=044 • 


"053 : 


1=062 




502 


70 070 


079 


088 


096 


i°5 


114 


122 


131 


140 


148 




503 


157 


165 


174 


183 


191 


200 


209 


217 


226 


234 




504 


243 


252 


260 


269 


278 


286 


295 


3°3 


312 


321 




505 


3 2 9 


338 


346 


355 


3 6 4 


372 


381 


389 


398 


406 




506 


415 


424 


432 


441 


449 


458 


467 


475 


484 


49 2 




507 


5 01 


5°9 


5i8 


526 


535 


544 


552 


561 


5 6 9 


578 




508 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 


I 


9 

d. 9 


509 


672 


680 


6S9 


697 


706 


714 


723 


73i 


740 


749 


2 


1.8 


510 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 


3 


2.7 

7 fi 


5" 


842 


851 


859 


868 


876 


885 


893 


902 


910 


919 


4 
5 


3-° 
45 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


+003 


6 

7 
8 


5-4 
6.3 
7.2 


5i3 


71 012 


020 


029 


°37 


046 


054 


063 


071 


079 


088 


5i4 


096 


105 


ii3 


122 


130 


139 


147 


155 


164 


172 


9 


8.1 


5i5 


181 


189 


198 


206 


214 


223 


231 


240 


248 


257 




5i6 


265 


273 


282 


290 


299 


3°7 


3i5 


324 


332 


34i 




517 


349 


357 


366 


374 


383 


39i 


399 


408 


416 


425 




5i8 


433 


441 


45° 


458 


466 


475 


483 


492 


500 


5°8 




519 


5i7 


525 


533 


542 


55° 


559 


5 6 7 


575 


584 


59 2 




520 


600 


609 


617 


625 


634 


642 


650 


659 


667 


675 




521 


684 


692 


700 


709 


717 


725 


734 


742 


75° 


759 




g 


522 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


1 


0.8 


523 


850 


858 


867 


875 


883 


892 


900 


908 


917 


9 2 5 


2 


1.6 


524 


933 


941 


95° 


958 


966 


975 


983 


991 


999 


*oo8 


3 

4 


2.4 

3- 2 


525 


72 016 


024 


032 


041 


049 


°57 


066 


074 


082 


090 


5 


4.0 


526 


099 


107 


ii5 


123 


132 


140 


148 


156 


165 


173 


6 

7 


4.8 

5.6 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 


255 


8 


6.4 


528 


263 


272 


280 


288 


296 


3°4 


3*3 


321 


329 


337 


9 


7-2 


S29 


346 


354 


362 


37° 


378 


387 


395 


403 


411 


419 




530 


428 


43 6 


444 


452 


460 


469 


477 


485 


493 


5oi 




53i 


5°9 


5i8 


526 


534 


542 


55° 


558 


567 


575 


583 




532 


S9i 


599 


607 


616 


624 


632 


640 


648 


656 


665 




533 


673 


681 


689 


697 


7°5 


7i3 


722 


73° 


738 


746 




534 


754 


76a 


770 


779 


787 


795 


803 


811 


819 


827 




535 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 




7 

0.7 


536 


916 


925 


933 


941 


949 


957 


965 


973 


981 


989 


1 


537 


997 


*oo6 


*oi4 


*02 2 


+030 


*o38 


^046 


*054 


*o62 


*o7o 


2 


1.4 


538 


73 °7 8 


086 


094 


I02 


in 


119 


127 


135 


143 


151 


3 

4 


2 !8 


539 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 


5 
6 

7 


3-5 


540 


239 


247 


255 


263 


272 


280 


288 


296 


3°4 


312 


4. 2 
4.9 


541 


320 


328 


33 6 


344 


352 


360 


368 


37 6 


384 


392 


8 


5-6 


542 


400 


408 


416 


424 


43 2 


440 


448 


45 6 


464 


472 


9 


6.3 


543 


480 


488 


496 


5°4 


512 


520 


528 


536 


544 


552 




544 


560 


568 


57 6 


584 


592 


600 


608 


616 


624 


632 




545 


640 


648 


656 


664 


672 


679 


687 


695 


7°3 


711 




546 


719 


727 


735 


743 


75i 


759 


767 


775 


783 


791 




547 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 




548 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 




549 


957 


965 


973 


981 


989 


997 


*oo5 


*°i3 


*020 


*028 






74 





















160 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


550 


74 °3 6 


044 


052 


060 


068 


076 


084 


092 


099 


107 




551 


US 


123 


131 


139 


147 


155 


162 


170 


178 


186 




552 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 




553 


273 


280 


288 


296 


3°4 


312 


320 


3 2 7 


335 


343 




554 


35i 


359 


367 


374 


382 


39° 


398 


406 


414 


421 




555 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 




556 


5°7 


5i5 


5 2 3 


53i 


539 


547 


554 


562 


57° 


578 




557 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 




558 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 




559 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 




560 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 




56i 


896 


904 


912 


920 


927 


935 


943 


95° 


958 


966 




562 


974 


981 


989 


997 


*oo5 


*OI2 


*020 


*028 


*°35 


*043 




8 

8 


563 


75 051 


°59 


066 


074 


082 


089 


097 


i°5 


113 


120 


I 


564 


128 


136 


i43 


151 


159 


l66 


174 


182 


189 


197 


2 


1 


6 


565 


205 


213 


220 


228 


236 


2 43 


2 5i 


2 59 


266 


274 


3 

4 


2 
3 


4 
2 


566 


282 


289 


297 


3°5 


312 


320 


328 


335 


343 


35i 


5 


4 





567 


358 


366 


374 


381 


389 


397 


404 


412 


420 


427 


6 

7 
8 


4 

s 

6 


8 
6 


568 


435 


442 


45° 


458 


465 


473 


481 


488 


496 


5°4 


4 


569 


5ii 


5i9 


526 


534 


54 2 


549 


.557 


565 


572 


580 


9 


7 


2 


570 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 




57i 


664 


671 


679 


686 


694 


702 


709 


7i7 


724 


732 




572 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 




573 


8i5 


823 


831 


838 


846 


853 


861 


868 


876 


884 




574 


891 


899 


906 


914 


921 


929 


937 


944 


95 2 


959 




575 


967 


974 


982 


989 


997 


*oo5 


*OI2 


*020 


*027 


*°35 




576 


76 042 


050 


°57 


065 


072 


080 


087 


°95 


103 


no 




577 


118 


125 


i33 


140 


148 


155 


163 


170 


178 


185 




578 


193 


200 


208 


215 


223 


230 


238 


2 45 


253 


260 




579 


268 


275 


283 


290 


298 


3°5 


313 


320 


328 


335 




580 


343 


35° 


358 


365 


373 


380 


388 


395 


403 


410 






58i 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 


1 


7 

0.7 


582 


492 


500 


5°7 


5i5 


522 


53° 


537 


545 


55 2 


559 


2 


1 


4 


583 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 


3 

4 


2 
2 


1 
8 


584 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 


5 


3 


5 


585 


716 


7 2 3 


73° 


738 


745 


753 


760 


768 


775 


782 


6 
7 
8 


4 


2 


586 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 


4 
5 


9 
6 


587 


864 


871 


879 


886 


893 


901 


908 


916 


9 2 3 


93° 


9 





3 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 


*oo4 




589 


77 012 


019 


026 


°34 


041 


048 


056 


063 


070 


078 




590 


085 


093 


100 


107 


ii5 


122 


129 


137 


144 


.^ 




59i 


159 


166 


i73 


181 


188 


195 


203 


210 


217 


225 




592 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 




593 


3°5 


313 


320 


3 2 7 


335 


34 2 


349 


357 


3 6 4 


37i 




594 


379 


386 


393 


401 


408 


415 


422 


43° 


437 


444 




595 


45 2 


459 


466 


474 


481 


488 


495 


5°3 


5i° 


517 




596 


525 


532 


539 


546 


554 


56i 


568 


576 


583 


59° 




597 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 




598 


670 


677 


685 


692 


699 


706 


7i4 


721 


728 


735 




599 


743 


75° 


757 


764 


772 


779 


786 


793 


801 


808 





TABLES 
Table XIV. Continued. — Logarithms of Numbers 



161 



No. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


600 


77 8l 5 


822 


830 


837 


844 


851 


859 


866 


873 


880 




601 


887 


895 


902 


909 


916 


924 


93i 


938 


945 


952 




602 


960 


967 


974 


981 


988 


996 


*oo3 


*OIO 


*oi7 


*025 




603 


78 032 


°39 


046 


°53 


061 


068 


°75 


082 


089 


O97 




604 


104 


in 


118 


125 


132 


140 


147 


154 


161 


168 




605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


24O 




606 


247 


254 


262 


269 


276 


283 


290 


297 


3°5 


312 




607 


319 


326 


333 


34° 


347 


355 


362 


369 


376 


383 




608 


39° 


398 


405 


412 


419 


426 


433 


440 


447 


455 




8 

s 


609 


462 


469 


476 


483 


490 


497 


5°4 


512 


5i9 


526 


2 


1 


6 


6lO 


533 


54o 


547 


554 


561 


569 


576 


583 


59° 


597 


3 


2 


4 


6ll 


604 


611 


618 


625 


6 33 


640 


647 


654 


661 


668 


4 

5 


3 
4 


2 



6l2 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 


6 


4 


8 


613 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 


7 
8 


S 
6 


6 
4 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


9 


7 


2 


615 


888 


895 


902 


909 


916 


923 


93° 


937 


944 


95i 




616 


958 


965 


972 


979 


986 


993 


*ooo 


*oo7 


*oi4 


*02I 




617 


79 029 


c 3 6 


043 


050 


°57 


064 


071 


078 


085 


O92 




618 


099 


106 


"3 


120 


127 


134 


141 


148 


155 


l62 




619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 




62O 


239 


246 


253 


260 


' 267 


274 


281 


288 


295 


302 




621 


3°9 


316 


323 


33° 


337 


344 


351 


358 


365 


372 




622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


1 


/ 

O *7 


623 


449 


45 6 


463 


470 


477 


484 


49 1 


498 


5°5 


5" 


2 


I 


4 


624 


5i8 


525 


532 


539 


546 


553 


560 


567 


574 


58l 


3 

4 


2 
2 


1 
8 


625 


588 


595 


602 


609 


616 


623 


630 


637 


644 


650 


5 


3 


S 


626 


6 57 


664 


671 


678 


685 


692 


699 


706 


713 


720 


6 


4 


2 


627 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 


7 
8 


4 

5 


9 
6 


628 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 


9 


6 


3 


629 


865 


872 


879 


886 


893 


900 


906 


9 J 3 


920 


927 




630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 




631 


80 003 


010 


017 


024 


030 


°37 


044 


051 


058 


065 




632 


072 


079 


085 


092 


099 


106 


"3 


120 


127 


134 




633 


140 


147 


i54 


161 


168 


i75 


182 


188 


i95 


202 




634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 




635 


277 


284 


291 


298 


3°5 


312 


318 


3 2 5 


33 2 


339 




5 


636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


I 


0.6 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


2 


1 


2 


638 


482 


489 


49 6 


502 


5°9 


5i6 


523 


53° 


536 


543 


3 

4 


1 
2 


8 
4 


639 


55° 


557 


564 


57° 


577 


584 


59i 


598 


604 


611 


5 


3 





640 


618 


625 


632 


63S 


645 


652 


659 


665 


672 


679 


6 
7 

8 


3 


6 


641 


686 


693 


699 


706 


7i3 


720 


726 


733 


740 


747 


4 


8 


642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


9 


5-4 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 




644 


889 


895 


902 


909 


916 


922 


929 


93 6 


943 


949 




645 


95 6 


963 


969 


976 


983 


990 


996 


*oo3 


*OIO 


*oi7 




646 


81 023 


030 


°37 


043 


050 


o57 


064 


070 


077 


084 




647 


090 


097 


104 


in 


117 


124 


131 


137 


144 


151 




648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 




649 


224 


231 


238 


245 


251. 


258 


265 


271 


278 


28^ 





11 



162 COMPRESSED AIR 

Table XIV. Continued. — Logarithms or Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


650 


81 291 


298 


3°5 


3ii 


318 


3 2 5 


33i 


338 


345 


35i 




651 


358 


365 


37i 


378 


385 


39i 


398 


405 


411 


418 




652 


425 


43i 


43 8 


445 


45i 


458 


465 


47i 


478 


485 




653 


491 


498 


5o5 


5" 


5i8 


525 


53i 


538 


544 


55i 




654 


558 


564 


57i 


578 


584 


59i 


59 8 


604 


611 


617 




655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 




656 


690 


697 


704 


710 


717 


723 


73° 


737 


743 


75° 




657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




659 


889 


895 


902 


908 


9i5 


921 


928 


935 


941 


948 




660 


954 


961 


968 


974 


981 


987 


994 


+000 


*oo7 


*oi4 




66l 


82 020 


027 


°33 


040 


046 


°53 


060 


066 


°73 


079 




662 


086 


092 


099 


i°5 


112 


119 


125 


132 


138 


145 






663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


1 


7 
1 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


2 


1 


4 


665 


282 


289 


295 


302 


308 


3i5 


321 


328 


334 


341 


3 

4 


2 
2 


1 
8 


666 


347 


354 


360 


3 6 7 


373 


380 


387 


393 


400 


406 


5 


3 


5 


667 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 


6 


4 


2 


668 


478 


484 


49 1 


497 


5°4 


5i° 


5i7 


523 


53° 


536 


8 


5 


6 


669 


543 


549 


556 


562 


5 6 9 


575 


582 


588 


595 


601 


9 


6 


3 


670 


607 


614 


620 


627 


6 33 


640 


646 


6 53 


659 


666 




671 


672 


679 


685 


692 


.698 


7o5 


711 


718 


724 


73° 




672 


737 


743 


75° 


756 


763 


769 


776 


782 


789 


795 




673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 




674 


866 


872 


879 


88 5 


892 


898 


9°5 


911 


918 


924 




675 


93° 


937 


943 


95° 


95 6 


963 


969 


975 


982 


988 




676 


995 


*OOI 


*oo8 


*oi4 


*020 


*02 7 


*°33 


+040 


=•=046 


*052 




677 


83 o59 


065 


072 


078 


08 5 


091 


097 


104 


no 


117 




678 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 




679 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 




680 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 




e. 


681 


3i5 


321 


327 


334 


340 


347 


353 


359 


366 


372 


1 


0.6 


682 


378 


385 


39i 


398 


404 


410 


417 


423 


429 


43 6 


2 


1 


2 


683 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 


3 

4 


1 
2 


8 
4 


684 


506 


512 


5i8 


525 


531 


537 


544 


55° 


556 


5 6 3 


5 


3 





685 


5 6 9 


575 


582 


588 


594 


601 


607 


613 


620 


626 


6 
7 
8 


3 


6 


68'6 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 


4 
4 


8 


687 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 


9 


5 


• 4 


688 


759 


765 


771 


778 


784 


790 


797 


803 


809 


816 




689 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 




690 


88 S 


891 


897 


904 


910 


916 


923 


929 


935 


942 




691 


948 


954 


960 


967 


973 


979 


985 


992 


998 


*oo4 




692 


84 on 


017 


023 


029 


036 


042 


048 


o55 


061 


067 




693 


°73 


080 


086 


092 


098 


i°5 


in 


117 


123 


130 




694 


136 


142 


148 


155 


161 


167 


i73 


180 


186 


192 




695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 




696 


261 


267 


273 


280 


286 


292 


298 


3°5 


311 


317 




697 


323 


33° 


33 6 


342 


348 


354 


361 


3 6 7 


373 


379 




698 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 




699 


448 


454 


460 


466 


473 


479 


485 


491 


497 


5°4 





TABLES 
Table XIV. Continued. — Logarithms of Numbers 



163 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pt8. 


700 


84 510 


516 


522 


528 


535 


54i 


547 


553 


559 


566 




701 


572 


578 


584 


59° 


597 


603 


609 


615 


621 


628 




702 


. 634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


75i 




704 


i 757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




705 


j 819 


825 


831 


837 


844 


850 


856 


862 


868 


874 




706 


880 


887 


893 


899 


9°5 


911 


917 


924 


93° 


936 




707 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 




708 


85 003 


009 


Ol6 


022 


028 


034 


040 


046 


052 


058 


I 


7 


709 


065 


071 


077 


083 


089 


o95 


IOI 


107 


114 


120 


2 


1 


4 


710 


126 


132 


138 


144 


i5° 


156 


163 


169 


175 


181 


3 


2 


1 
8 
S 


711 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


4 
5 


3 


712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


3°3 


6 


4 


2 


713 


3°9 


315 


321 


327 


333 


339 


345 


352 


358 


3 6 4 


7 
8 


4 

5 


9 
6 


714 


37° 


376 


382 


388 


394 


400 


406 


412 


418 


425 


9 


6 


3 


715 


43i 


437 


443 


449 


455 


461 


467 


473 


479 


485 




716 


491 


497 


5°3 


5°9 


5i6 


522 


528 


534 


54o 


546 




717 


552 


558 


564 


57° 


576 


582 


588 


594 


600 


606 




718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 




719 


673 


679 


685 


691 


697 


7°3 


709 


7i5 


721 


727 




720 


733 


739 


745 


75i 


757 


763 


769 


775 


781 


788 




721 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 




722 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


1 




fi 


723 


914 


920 


926 


932 


938 


944 


95° 


95 6 


962 


968 


2 


1 


2 


724 


974 


980 


986 


992 


998 


*oo4 


*OIO 


*oi6 


*022 


*028 


3 

4 

5 


1 


8 
4 



725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


3 


726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 


6 
7 

8 


3 


6 


727 


153 


iS9 


165 


171 


177 


183 


189 


195 


201 


207 


4 
4 


8 


728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


9 


5 


4 


729 


273 


279 


285 


291 


297 


3°3 


308 


3i4 


320 


326 




730 


332 


338 


344 


35° 


356 


362 


368 


374 


380 


386 




731 


392 


398 


404 


410 


4i5 


421 


427 


433 


439 


445 




732 


45i 


457 


463 


469 


475 


481 


487 


493 


499 


5°4 




733 


Sio 


5i6 


522 


528 


534 


54o 


546 


552 


558 


564 




734 


57° 


576 


58i 


587 


593 


599 


605 


611 


617 


623 




735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 






736 


688 


694 


700 


7°5 


711 


717 


723 


729 


735 


741 


1 


5 

0.5 


737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


2 


1 





738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


3 

4 


1 
2 


5 



739 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


5 


2 


S 


740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 


6 


3 



5 



74i 


; 982 


988 


994 


999 


*oc>5 


*OII 


*oi7 


*023 


*029 


*°35 


8 


4 


742 


87 040 


046 


052 


058 


064 


070 


°75 


081 


087 


°93 


9 


4-5 


743 


099 


i°5 


in 


116 


122 


128 


134 


140 


146 


151 




744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 




745 


216 


221 


227 


•233 


239 


245 


251 


256 


262 


268 




746 


274 


280 


286 


291 


297 


3°3 


3°9 


315 


320 


326 




747 


332 


338 


344 


349 


355 


361 


3 6 7 


373 


379 


384 




748' 


39° 


396 


402 


408 


4i3 


419 


425 


43i 


437 


442 




749 


448 


454 


460 


466 


47i 


477 


483 


489 


495 


500 





164 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


750 


87 506 


512 


5i8 


523 


529 


535 


54i 


547 


552 


558 




751 


564 


57° 


576 


58i 


587 


593 


599 


604 


610 


616 




752 


622 


628 


6 33 


639 


645 


651 


656 


662 


668 


674 




753 


679 


685 


691 


697 


7°3 


708 


714 


720 


726 


73i 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 




756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


9i5 


921 


927 


933 


938 


944 


95° 


955 


961 




758 


967 


973 


978 


984 


990 


996 


*OOI 


*oo7 


*oi3 


*oi8 




759 


88 024 


030 


036 


041 


047 


°53 


058 


064 


070 


076 




760 


081 


087 


°93 


098 


104 


no 


116 


121 


127 


133 




761 


138 


144 


150 


156 


161 


167 


173 


178 


184 


190 




762 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 




6 

0.6 


763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


3°4 


1 


764 


3°9 


3i5 


321 


326 


332 


338 


343 


349 


355 


360 


2 


1 


2 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


4i7 


3 

4 


1 
2 


8 
4 


766 


423 


429 


434 


440 


446 


45i 


457 


463 


468 


474 


5 


3 





767 


480 


485 


49 1 


497 


502 


508 


5i3 


519 


525 


53° 


6 


3 


6 


768 


536 


542 


547 


553 


559 


564 


57° 


576 


581 


587 


7 

8 


4 
4 


8 


769 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 


9 


5 


4 


770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




771 


705 


711 


717 


722 


728 


734 


739 


745 


75o 


756 




772 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 




775 


93o 


93 6 


941 


947 


953 


958 


964 


969 


975 


981 




776 


986 


992 


997 


*oo3 


*oo9 


*oi4 


*020 


*025 


*03i 


*Q37 




777 


89 042 


048 


°53 


o59 


064 


070 


076 


081 


087 


092 




778 


098 


104 


109 


ii5 


120 


126 


131 


137 


143 


148 




779 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 




780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




781 


265 


271 


276 


282 


287 


293 


298 


3°4 


310 


3i5 


I 


5 

0. <: 


782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


37i 


2 


1 





783 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 


3 

4 


1 
2 


5 



784 


43 2 


437 


443 


448 


454 


459 


465 


470 


476 


481 


5 


2 


5 


785 


487 


492 


498 


5°4 


5°9 


515 


520 


526 


53i 


537 


6 


3 





786 


542 


548 


553 


559 


5 6 4 


57° 


575 


S81 


586 


592 


7 
8 


3 

4 


S 



787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


9 


4 


5 


788 


653 


658 


664 


669 


6 75 


680 


686 


691 


697 


702 




789 


708 


7i3 


•719 


724 


73° 


735 


741 


746 


752 


757 




790 


7 6 3 


768 


774 


779 


785 


790 


796 


801 


807 


812 




791 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




792 


873 


878 


883 


889 


894 


900 


9°5 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 


*oo4 


*oc>9 


=1=015 


*020 


*026 


*o 3 i 




795 


90 037 


042 


048 


o53 


059 


064 


069 


°75 


080 


086 




796 


091 


097 


102 


108 


"3 


119 


124 


I29 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




799 


255 


260 


266 


271 


276 


282 


287 


293 


298 


3°4 





TABLES 

Table XIV. Continued. — Logarithms of Numbers 



165 



No. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


800 


90 309 


314 


320 


325 


33* 


336 


342 


347 


352 


358 




801 


3 6 3 


369 


374 


380 


385 


39° 


396 


401 


407 


412 




802 


417 


423 


428 


434 


439 


445 


45° 


455 


461 


466 




803 


472 


477 


482 


488 


493 


499 


5°4 


5°9 


5i5 


520 




804 


526 


53i 


536 


542 


547 


553 


558 


5 6 3 


5 6 9 


574 




805 


580 


585 


59° 


596 


601 


607 


612 


617 


623 


628 




806 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 




807 


687 


693 


698 


7°3 


709 


714 


720 


725 


73° 


736 




808 


741 


747 


752 


757 


7 6 3 


768 


773 


779 


784 


789 




809 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




8lO 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 




8ll 


902 


907 


913 


918 


924 


929 


934 


940 


945 


95° 




8l2 


95 6 


961 


966 


972 


977 


982 


988 


993 


998 


*oo4 




5 


813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


°57 


1 


0.6 


814 


062 


068 


°73 


078 


084 


089 


094 


100 


i°5 


no 


2 


1 


2 


815 


116 


121 


126 


132 


t-37 


142 


148 


153 


158 


164 


3 
4 


1 
2 


8 
4 


8l6 


169 


i74 


180 


185 


190 


196 


201 


206 


212 


217 


5 


3 





817 


222 


228 


2 33 


238 


243 


249 


254 


259 


265 


270 


6 
7 


3 


6 


818 


275 


281 


286 


291 


297 


302 


3°7 


312 


318 


323 


8 


4 


8 


819 


328 


334 


339 


344 


35° 


355 


360 


365 


37i 


376 


9 


5 


4 


82O 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 




821 


434 


440 


445 


45° 


455 


461 


466 


47i 


477 


482 




822 


487 


492 


498 


5°3 


508 


5i4 


5i9 


524 


529 


535 




823 


54o 


545 


55i 


556 


56i 


566 


572 


577 


582 


587 




824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 




825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


6 93 




826 


698 


7°3 


709 


7i4 


719 


724 


73° 


735 


740 


745 




827 


75i 


756 


761 


766 


772 


777 


782 


787 


793 


798 




828 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 




829 


855 


861 


866 


871 


876 


882 


887 


892 


897 


9°3 




83O 


908 


913 


918 


924 


929 


934 


939 


944 


95° 


955 






831 


960 


965 


971 


976 


981 


986 


991 


997 


*002 


*oo7 


1 


5 
<; 


832 


92 012 


018 


023 


028 


o33 


038 


044 


049 


054 


o59 


2 


1 





833 


065 


070 


°75 


080 


085 


091 


096 


IOI 


106 


in 


3 

4 


1 
2 


S 



834 


117 


122 


127 


132 


i37 


143 


148 


153 


158 


163 


5 


2 


s 


835 


169 


i74 


179 


184 


189 


195 


200 


205 


2IO 


215 


6 
7 
8 


3 





836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


3 
4 


5 



837 


273 


278 


283 


288 


293 


298 


3°4 


3°9 


314 


3 J 9 


9 


4 


5 


838 


324 


33° 


335 


34o 


345 


35° 


355 


361 


366 


37i 




839 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 




84O 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 




841 


480 


485 


490 


495 


500 


5°5 


5ii 


5i6 


521 


526 




842 


53i 


536 


542 


547 


552 


557 


562 


567 


572 


578 




843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 




844 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 




845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




846 


737 


742 


747 


75 2 


758 


7 6 3 


768 


773 


778 


783 




847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 




848 


840 


845 


850 


855 


860 


865 


870 


875 


88l 


886 




849 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 





166 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


850 


92 942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




851 


993 


998 


*oo3 


*oo8 


*oi3 


*oi8 


*024 


*029 


*°34 


*039 




852 


93 °44 


049 


o54 


°59 


064 


069 


°75 


080 


085 


090 




853 


°95 


100 


i°5 


no 


ii5 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




855 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 




856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




857 


298 


3°3 


308 


3*3 


318 


323 


328 


334 


339 


344 




858 


349 


354 


359 


364 


3 6 9 


374 


379 


384 


389 


394 


1 


6 

i 


859 


399 


404 


409 


414 


420 


425 


43° 


435 


440 


445 


2 


1 


2 


860 


45° 


455 


460 


465 


470 


475 


480 


485 


490 


495 


3 


1 


8 


86l 


500 


5°5 


5i° 


515 


520 


526 


53i 


53 6 


541 


546 


4 

5 


2 
3 


4 



862 


551 


556 


561 


566 


57i 


576 


58i 


586 


59i 


596 


6 


3 


6 


863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 


7 

8 


4 


2 

8 


864 


6Si 


656 


661 


666 


671 


676 


682 


687 


692 


697 


9 


5 


4 


865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


95 2 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 


94 002 


007 


012 


017 


022 


027 


032 


°37 


042 


047 




872 


052 


°57 


062 


067 


072 


077 


082 


086 


091 


096 




5 


873 


IOI 


106 


in 


116 


121 


126 


131 


136 


141 


146 


2 


1 





874 


151 


1S6 


161 


166 


171 


176 


181 


186 


191 


196 


3 

4 
5 


1 


5 


875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 


2 


5 


876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 


6 

7 
8 


3 





877 


300 


3°5 


310 


3i5 


320 


325 


33° 


335 


34o 


345 


3- 

4 


5 



878 


349 


354 


359 


3 6 4 


369 


374 


379 


384 


389 


394 


9 


4 


5 


879 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 




880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 




881 


498 


5°3 


5°7 


512 


5i7 


522 


527 


532 


537 


542 




882 


547 


552 


557 


562 


5 6 7 


57i 


576 


58i 


586 


591 




883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




884 


64S 


650 


655 


660 


665 


670 


675 


680 


685 


689 




885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 






886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 


1 


4 
0.4 


887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


2 





8 


888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 


3 

4 


1 
1 


2 
6 


889 


890 


895 


900 


9°5 


910 


9i5 


919 


924 


929 


934 


5 


2 





890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 


6 

7 
8 


2 


4 
8 


891 


988 


993 


998 


*002 


*oo7 


*OI2 


*oi7 


*02 2 


*02 7 


^032 


3 


2 


892 


95 36 


041 


046 


05I 


056 


061 


066 


071 


075 


080 


9 


3-6 


893 


085 


090 


°95 


IOO 


i°5 


IO9 


114 


119 


124 


129 




894 


i34 


139 


143 


-I48 


153 


158 


163 


168 


173 


177 




895 


182 


187 


192 


197 


202 


207 


211 


2l6 


221 


226 




896 


231 


236 


240 


245 


250 


255 


260 


265 


270 


274 




897 


279 


284 


289 


294 


299 


3°3 


308 


313 


318 


323 




898 


328 


33 2 


337 


342 


347 


352 


357 


361 


366 


37i 




899 


376 


381 


386 


39° 


395 


400 


405 


4IO 


415 


419 





TABLES 
Table XIV. Continued. — Logarithms op Numbers 



167 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


r 

Pp. Pts. 


900 


95 424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




901 


472 


477 


482 


487 


492 


497 


5°i 


506 


5ii 


5i6 




902 


5 2 i 


525 


53° 


535 


540 


545 


55° 


554 


559 


5 6 4 




903 


S69 


574 


578 


583 


588 


593 


598 


602 


607 


612 




904 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 




905 


665 


670 


674 


679 


684 


689 


694 


698 


7°3 


708 




906 


713 


718 


722 


727 


73 2 


737 


742 


746 


75i 


756 




907 


761 


766 


770 


775 


780 


785 


789 


794 


799 


804 




908 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 




909 


856 


861 


866 


871 


875 


880 


885 


890 


895 


899 




910 


904 


909 


914 


918 


923 


928 


933 


938 


943 


947 




911 


952 


957 


961 


966 


971 


976 


980 


985 


990 


995 




912 


999 


*oo4 


*oo9 


*oi4 


*oi9 


*023 


*028 


*°33 


*o 3 8 


#042 




5 
0.5 


913 


96 047 


052 


057 


061 


066 


071 


076 


080 


085 


090 


1 


914 


095 


099 


104 


109 


114 


118 


123 


128 


133 


i37 


2 


1 





915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


3 

4 


1 
2 


5 



916 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


5 


2 


5 


917 


237 


242 


246 


2 5 J 


256 


261 


265 


270 


275 


280 


6 
7 


3 



5 


918 


284 


289 


294 


298 


3°3 


308 


3*3 


3i7 


322 


327 


8 


4 





919 


332 


33 6 


34i 


346 


35° 


355 


360 


3 6 5 


369 


374 


9 


4-5 


920 


379 


384 


388 


393 


398 


402 


407 


412 


4i7 


421 




921 


426 


43i 


435 


440 


445 


45° 


454 


459 


464 


468 




922 


473 


478 


483 


487 


49 2 


497 


5°i 


506 


5ii 


5i5 




923 


520 


525 


53° 


534 


539 


544 


548 


553 


558 


562 




924 


567 


572 


577 


581 


586 


59i 


595 


600 


605 


609 




925 


614 


619 


624 


628 


£>33 


638 


642 


647 


652 


656 




926 


661 


666 


670 


675 


680 


685 


689 


694 


699 


7°3 




927 


708 


7i3 


717 


722 


727 


73i 


736 


741 


745 


75° 




928 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 




929 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 




930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 




4 

4 


931 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 


1 


932 


942 


946 


95i 


956 


960 


965 


970 


974 


979 


984 


2 





8 


933 


988 


993 


997 


*002 


*oc>7 


*OII 


*oi6 


*02I 


*025 


*o3o 


3 

4 


1 
1 


2 
6 


934 


97 °35 


°39 


044 


O49 


°53 


058 


063 


067 


072 


077 


5 


2 





935 


081 


086 


090 


°95 


100 


104 


109 


114 


118 


123 


6 
7 

8 


2 


4 
8 


936 


128 


132 


137 


142 


146 


151 


155 


l6o 


165 


169 


3 


2 


937 


174 


179 


183 


l88 


192 


197 


202 


206 


211 


216 


9 


3 6 


938 


220 


225 


230 


234 


239 


243 


248 


253 


257 


262 




939 


267 


271 


276 


280 


285 


290 


294 


299 


3°4 


308 




940 


3*3 


3i7 


322 


327 


33i 


336 


34o 


345 


35° 


354 




941 


359 


3 6 4 


368 


373 


377 


382 


387 


39i 


396 


400 




942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




943 


45i 


45 6 


460 


465 


470 


474 


479 


483 


488 


493 




944 


497 


5°2 


506 


5" 


5i6 


520 


525 


529 


534 


539 




945 


543 


548 


552 


557 


562 


566 


57i 


575 


580 


585 




946 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 




947 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 




948 


681 


685 


690 


695 


699 


704 


708 


7i3 


717 


722 




949 


727 


73i 


736 


740 


745 


749 


754 


759 7 6 3 


768 





168 COMPRESSED AIR 

Table XIV. Continued. — Logarithms of Numbers 



Ko. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


950 


97 772 


777 


782 


786 


791 


795 


800 


804 


809 


8i3 




9Si 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 




952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


90S 




953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


95° 




954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 




955 


98 000 


005 


009 


014 


019 


023 


028 


032 


°37 


041 




956 


046 


050 


°55 


°59 


064 


068 


073 


078 


082 


087 




957 


091 


096 


100 


i°5 


109 


114 


118 


123 


127 


132 




958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 




959 


182 


186 


191 


i95 


200 


204 


209 


214 


218 


223 




960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 




961 


272 


277 


281 


286 


290 


295 


299 


3°4 


308 


3i3 




962 


3i8 


322 


327 


33i 


336 


34o 


345 


349 


354 


358 






963 


3 6 3 


367 


372 


376 


381 


385 


39° 


394 


399 


403 


1 


5 

O- c 


964 


408 


412 


417 


421 


426 


43° 


435 


439 


444 


448 


2 


I 





965 


453 


457 


462 


466 


47i 


475 


480 


484 


489 


493 


3 

4 


I 

2 


5 



966 


498 


502 


507 


5ii 


5i6 


520 


525 


529 


534 


538 


5 


2 


5 


967 


543 


547 


55 2 


556 


56i 


565 


57° 


574 


579 


583 


6 

7 
8 


3 





968 


588 


592 


597 


601 


605 


610 


614 


619 


623- 


628 


3 

4 


5 



969 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


9 


4 


5 


970 


677 


682 


686 


691 


695 


700 


704 


709 


7i3 


717 




971 


722 


726 


73i 


735 


740 


744 


749 


753 


758 


762 




972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 




973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 




974 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 




975 


900 


9°5 


909 


914 


918 


923 


927 


93 2 


936 


941 




976 


945 


949 


954 


958 


9 6 3 


967 


972 


976 


981 


985 




977 


989 


994 


998 


*oo3 


*oo7 


*OI2 


*oi6 


*02I 


*025 


*029 




978 


99 °34 


038 


043 


047 


052 


056 


061 


065 


069 


074 




979 


078 


083 


087 


092 


096 


IOO 


i°5 


IO9 


114 


118 




980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




981 


167 


171 


176 


180 


185 


189 


193 


I98 


202 


207 


1 


4 

O A 


982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 


2 


O 


8 


983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


3 

4 


I 
I 


2 
6 


984 


300 


3°4 


308 


313 


3i7 


322 


326 


33° 


335 


339 


5 


2 





985 


344 


348 


352 


357 


361 


366 


37° 


374 


379 


383 


6 


2 


4 
8 
2 


986 


388 


392 


396 


401 


405 


4IO 


414 


419 


423 


427 


7 
8 


3 


987 


43 2 


43 6 


441 


445 


449 


454 


458 


463 


467 


47i 


9 


3 


6 


988 


476 


480 


484 


489 


493 


498 


502 


506 


5ii 


5i5 




989 


520 


524 


528 


533 


537 


542 


546 


55° 


555 


559 




990 


5 6 4 


568 


572 


577 


58i 


585 


59° 


594 


599 


603 




991 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 




992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 




993 


6 95 


699 


704 


708 


712 


717 


721 


726 


739 


734 




994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 




995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 




996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


86 S 




997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




998 


9i3 


917 


922 


926 


93° 


935 


939 


944 


948 


952 




999 


957 


961 


96S 


970 


974 


978 


983 


987 


991 


996 





APPENDIX A 

The following notes and tables relating to drill capacities are 
taken from the Ingersoll-Rand catalog. 

DRILL CAPACITY TABLES 

The following tables are to determine the amount of free air 
required to operate rock drills at various altitudes with air at 
given pressures. 

The tables have been compiled from a review of a wide expe- 
rience and from tests run on drills of various sizes. They are in- 
tended for fair conditions in ordinary hard rock, but owing to 
varying conditions it is impossible to make any guarantee with- 
out a full knowledge of existing conditions. 

In soft material where the actual time of drilling is short, more 
drills can be run with a given sized compressor than when working 
in hard material, when the drills would be working continuously 
for a longer period, thereby increasing the chance of all the drills 
operating at the same time. 

In tunnel work, where the rock is hard, it has been the expe- 
rience that more rapid progress has been made when the drills 
were operated under a high air pressure, and that it has been 
found profitable to provide compressor capacity in excess of the 
requirements by about 25 per cent. There is also a distinct ad- 
vantage in having a compressor of large capacity, in that it saves 
the trouble and expense of moving the compressor as the work 
progresses, and will not interfere with the progress of the work by 
crowding the tunnel. 

No allowance has been made in the tables for loss due to leaky 
pipes, or for transmission loss due to friction, but the capacities 
given are merely the displacement required, so that when select- 
ing a compressor for the work required these matters must be 
taken into account. 

Table I gives cubic feet of free air required to operate one drill 
of a given size and under a given pressure. 

Table II gives multiplication factors for altitudes and number 

169 



170 



COMPRESSED AIR 



of drills by which the air consumption of one drill must be multi- 
plied in order to give the total amount of air. 

Table 1.- — Cubic Feet of Free Air Required to Run One Drill of the 
Size and at the Pressure Stated Below 



u 

* § 

cS 
O 

60 
70 
80 
90 
100 


SIZE AND CYLINDER DIAMETER OF DRILL 


A35 

2" 

50 
56 
63 
70 

77 


A32 
2\" 

60 
68 
76 

84 
92 


B 

68 
77 
86 
95 
104 


C 
2f" 

82 

93 

104 

115 

126 


D 
3" 

90 
102 
114 
126 
138 


D 

ol " 

95 
108 
120 
133 
146 


D 

o 3 II 
°16 

97 
110 
123 
136 
149 


E 

si" 


F 
3|" 

108 
124 
131 
152 
166 


F 

a s 

113 
129 
143 
159 
174 


G 
4|" 

130 
147 
164 
182 
199 


H 

5" 

150 
170 
190 
210 
240 


H9 

6*"" 

164 
181 
207 
230 
252 


100 
113 
127 

141 
154 



GLOBE VALVES, TEES AND ELBOWS 

The reduction of pressure produced by globe valves is the same 
as that caused by the following additional lengths of straight pipe, 
as calculated by the formula: 



Additional length of pipe = 



114 X diameter of pipe 



1 -f- (36 -T- diameter) 

Diameter of pipe 1 1 1^ 2 2^ 3 3M 4 5 6 inches 

Additional length J 2 4 7 10 13 16 20 28 36 feet 

7 8 10 12 15 18 20 22 24 inches 

44 53 70 88 115 143 162 181 200 feet 

The reduction of pressure produced by elbows and tees is equal 
to two-thirds of that caused by globe valves. The following are 
the additional lengths of straight pipe to be taken into account for 
elbows and tees. For globe valves multiply by ^. 



Diameter of pipe \ 
Additional length J 



1 W 2 2 2V 2 3 3K 

2 3 5 7 9 11 
7 8 10 12 15 18 

30 35 47 59 77 96 



4 5 6 inches 

13 19 24 feet 

20 22 24 inches 

108 120 134 feet 



These additional lengths of pipe for globe valves, elbows and 
tees must be added in each case to the actual lengths of straight 
pipe. Thus, a 6-inch pipe, 500 ft. long, with 1 glove valve, 2 
elbows and 3 tees, would be equivalent to a straight pipe 500 + 
36 + (2 X 24) + (3 X 24) = 656 feet long. 



APPENDIX A 



171 



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APPENDIX B 
DESIGN OF LOGARITHMIC COMPUTING CHARTS 

Problem. — Design a chart for determining values of x, y and z 
in equations of the form: 

x n = ay m z r 



or 



whence 



1 m r^ 

x = a n y n z n 



i TYl V 

logo; = - log a + - logy + - log z (I) 

IV IV IV 

As a preliminary and introductory study assume n — m + r 
and construct a chart as follows (See Fig. 27) : 

Tabulate values of x, y and z to be covered or included in the 
chart. Take out the logarithms of these numbers. Plat these 
logarithms, to some convenient scale, on the vertical lines marked 
x, y and z setting the zero of scale at A and B for the y and z 
lines respectively, but for the x line set the zero of scale at F 

making FG = - log a. On the lines x, y and z mark the num- 
bers whose logs have been scaled. Then evidently wherever the 
line CD may be placed across the chart the proportions thereon 
written will hold and Eq. (I) is completely satisfied' — that is — 
given any two of x, y and z the third will be found on the line 
CD laid over the two given. 

Note that the line AE is unnecessary — it being placed in the 
figure for demonstration only. Note also that the line FG is not 
to appear on the chart and that the factor C effects only the width 
of the chart and may be taken to suit convenience. 

Evidently this solution applies only to the special case when 
n = m + r. It has the further objection that if the corre- 
sponding numerical values of x, y and z are very different then 
the lengths of the x, y and z lines will be different, though not 
in the same proprtion. The chart will have a better appearance 
if the three lines are nearly equal. 

The general solution is as follows (See Fig. 28) : 

172 



APPENDIX B 

Let I = desired length of chart in inches, 
k = desired width of chart in inches, 
Xi = greatest value of x to appear on the chart, 
f x be the necessary multiplier for logs x, 
to give desired length to the x line. 



173 




log z 



Then /*(log xi log a) = I Whence f x 

Let Z\ — be the value of z corresponding with X\. 



(II) 



174 



COMPRESSED AIR 



Let f z be the nearest whole number to 



I 



(HI) 



log Z\ 

that being the most convenient multiplier for 
logs of z to make the z line nearly equal to the 
x line. 




f z lo e z 



F-l 



Fig. 28. 



Let f y be the necessary multiplier of logs y. 

We have yet to find p, q and f y . 

Imposing the condition that Eq. (I) must be satisfied and 
remembering that all values of log x are to be multiplied by f x . 
Then must 



.APPENDIX B 175 

^f, log z = V f x log z. Whence p = -fy-k (IV) 

/c ft w/ z 

also 

| /.log y = | /, log y. Whence fv=^fh (V) 

Evidently q = k — p (VI) 

Example.- — -Design a chart to solve the formula for friction in 
air pipes, viz., 

0.1025 vH 



rd 5 - n X 3,600 
in which 

/ = loss of pressure in pounds per square inch, 

I = length of pipe in feet, 

v = cubic feet of free air per minute, 

r = ratio of compression = number of atmospheres, 

d = diameter of air pipe in inches. 

Here we find five variables while our chart can provide for three 
only. We will, therefore, take I = 1,000 feet and replace the 
product fr with a single variable and represent it by h. The equa- 
tion will now become 



Whence 



fr = h = 35J3 dP" ° r V ' = 35 ' 13 M5 ' 31 



log v = ^ log 35.13 + \ log h + -^ log d (VII) 



which is in the same form as Eq. I. 

We will design the chart to be about 12 in. long (I = 12) and 
8 in. wide (k = 8) and will provide for a Max. v = 50,000 (=vi) 
log 50,000 = 4.6990 and log 35.13 = 1.5456 
whence by II, 

1.5456\ 



/. (4.699 -^P)=12, 



whence /» = 3 (nearest whole number) . 

The value of d corresponding to v = 50,000 is about 12 in. 

12 
log 12 = 1.0792. Whence by III, f d = 1 Q?92 = 12 (nearest 

whole number). 
Then by IV, 

P = 5 f X8X^ = 5.31 and q = 8 - 531. = 2.69 



176 



COMPRESSED AIR 



and by V 



} - = 2 X 2^9 X 3 " 4Mh 



See Plate III, page 53, for the completed chart. 

To lay out such a chart, make a table such as is indicated below. 
Then measure out, on the respective lines, with a scale of inches 
and decimals (engineers scale) the quantities in columns headed 
f v log v, fh log h and f d log d remembering that log 1 = 0.0. 
Hence the bottom of each line A, F and B will be marked 1. As 
each multiplied log is marked on its line, write there the corre- 
sponding number in columns headed v, h and d respectively. 

On a chart thus laid out a thread stretched as at CD will lie 
over the three quantities that will satisfy the equation, hence any 
two being known the third can be found. 



Table for Charting Equation, v 2 = 35.13 hd 6 - 31 
Note Y% log 35.13 = 0.7728 and 3 X 0.7728 = 2.318 



v line 


h line 


d line 


V 


log V 


fv log V 


h 


log h 


fh log h 


d 


log d 


fd log d 







































The following notes may help the student when designing 
charts for other equations of the form given in Eq. I. 

(a) If the constant (a, Eq. I) becomes a proper fraction its log. 
is minus and the point F must be set above G instead of below; 
the zero of scale to be set at F when measuring out the X logs. 

(6) If the equation takes the form 



x n _ 



y m Z r 



then 



TYl T 

log x = - log a log y log z 

n n n 



APPENDIX B 



177 



This can be satisfied by reversing the direction of the measure- 
ments on the x line and placing the zero (or F) point above the 

line AB a distance - log af x as indicated in Fig. 29. 

it 

(c) When values of any one of the variables must be fractional 
the log is minus and must be measured in the opposite direction 
from A, B or F as the case may be. For instance if in the ex- 




Fig. 29. 



ample above d = % in. then log d = T.8751 = —0.1249 and we 
must set Y± at 0.1249/,* below B. 

(d) If circumstances are such that in the solution of such prob- 
lems as occur in practice; the figures on the lower portion of one 
of the lines (say the y line) will never be needed; the chart may 
be set out as suggested in Fig. 30 and only that portion above BR 
retained on the finished chart. Thus the scale may be enlarged 
and accuracy increased thereby. 



12 



178 



COMPRESSED AIR 



Evidently the essential proportions of Fig. 27 are not changed 
by putting the chart in the shape of Fig. 30. 



i 
i 
i 

1 |-F 
Fig. 30. 



(e) It will be found convenient to let x, in the above discus- 
sion, represent the largest factor in the equation. 



APPENDIX C 



During 1910 and 1911, an extensive series of experiments were 
made at Missouri School of Mines to determine the laws of fric- 
tion of air in pipes under three inches in diameter; the chief object 
being to determine the coefficient 



c" in the formula / = c -r 5 



(See Art. 29.) 



The general scheme is illustrated in Fig. 31, in which the parts 
are lettered as follows: 

a, is the compressed-air supply pipe. 

b, a receiver of about 25 cu. ft. capacity. 

c, a thermometer set in receiver. 

d and d, points of attachment of differential gage. 




Fig. 31. — Diagram illustrating assembled apparatus. 

/ and /, lengths of straight pipe going to and from the group of 
fittings. 

e, the pressure gage. 

g, the group of fittings — varied in different experiments. 

h, the throttle valve to control pressure. 

I, the orifice drum for measuring air, with the attachments as 
in Fig. 9. 

Experiments at Missouri School of Mines — 1911 

On each set of fittings there were made ten or twelve runs with 
varying pressures and quantities of air in order to show the rela- 

V 

tion of / to — over as wide a field as possible. 

179 



180 



COMPRESSED AIR 



£ e8 



II 


ft 


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APPENDIX G 181 

The data of each run were worked up and recorded in tabular 
form. Three of these tables, relating to 1-in. pipe and fittings, 
are shown herewith as example. It should be recorded that in the 
series of runs and checks some puzzling inconsistencies developed, 
but not more noticeable than appears in the data from European 
experiments on larger pipe. 

In these tables the symbols are as follows : 

z = head, in inches of mercury, in differential gage, ; 

/ = lost pressure in pounds per square inch, 
p 2 = gage pressure at entrance to pipe, 
r m = mean ratio of compression in pipe, 
i = water head, in inches, in U tube on orifice drum, 
T c = temperature (centigrade) in drum, 
d = diameter, in inches, of orifice in drum, 
v a = volume of free air passing (cubic feet per second), 

S = velocity of compressed air in pipe (feet per second), 
/' = value of /when corrected for temperature. 



182 



COMPRESSED AIR 



3 ^ 
2 w 

g 
tfl ft 





r^ 




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APPENDIX C 



183 



y 




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184 



COMPRESSED AIR 



Vn 

On platting the values of /and — as corresponding coordinates, 

it becomes apparent that they are related to each other in all cases 
as ordinates to a straight line ; which could have been anticipated 
from the established laws of fluid frictions. This is shown on 
Plate V. 





















\ ' - 






1 


















































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II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II 1 1 



















Values of "f"' 



From this plate we get the following three equations : 

80.0 K + 2e + 5u+4:g= 18.3, 
80.0 K + 10 e + 9u = 11.8, 

80.0 K + 2 e + 13 m = 6.8, 



in which 



APPENDIX C 185 



v 
K— = resistance due to one foot of pipe, 



v 
e — = resistance due to one elbow, 



r 



v 
m— = resistance due to one extra ferrule or joint 



r 



with ends reamed, 



v 
u— = resistance due to one extra ferrule or joint 



r 



g — = resistance due to one globe valve. 



with ends unreamed, 

2 

i 

r 

So by attaching other lengths or fittings we get other equations 
and by simple algebra can find the numerical value of each symbol. 

Then 

v j i y 2 

Kl — = c -= — — or c = d 5 K. 

r d b r 

Also the length of pipe giving friction equal to that of one elbow 
g 
is t, and so with other fittings. 

These experiments covered standard galvanized pipes of 2, 1}^, 
1, %, and 3^-inch diameter. With each size pipe, runs were made 
to find friction loss in ordinary elbows, 45° elbows, globe valves, 
return bends, unreamed joints, and reamed joints. For each 
combination, data were taken for platting twelve to eighteen 
points, altogether about eight hundred. The results as a whole 
are satisfactory for the 2-, IjHj-, and 1-inch pipes. 

For the %- and 3^-inch pipes, especially the 3^-in pip e ; the 
results were so irregular, erratic, and conflicting that the results 
finally recorded cannot be accepted as final. In the light of these 
results, it is not probable that a satisfactory coefficient will ever 
be gotten for pipes under 1 inch; the reason being that in pipes of 
such small diameter, irregularities have relatively much greater 
effect than in larger pipes, and the probability of obstructions 
lodging in such pipes is relatively greater. In the 3^-inch pipe 
and fitting, unreamed joints were found at which four-tenths of 
the area was obstructed, and this with a knife edge. No doubt 
consistent results could have been gotten by using only pipes 
that had been "plugged and reamed," and selected fittings, but 
these results would not have been a safe guide for practice unless 
such preparation of the pipe be specified. 



186 



COMPRESSED AIR 



The results of these researches are embodied in Plate II. They 
show the averages of such data as seem worthy of consideration. 
The data for pipes exceeding 2-in. diameter are taken from 
various published data. Verification of these by the use of the 
sensitive differential gage is desirable. 

In the series of experiments referred to, the results worked 
out for the resistance of fittings were more erratic than those for 
straight pipes. Hence no claim is made for precision or finality 
in the results here presented. However, two important conclu- 
sions are reached. One is that the resistance of globe valves has 
heretofore been underestimated, and the importance of reaming 
small pipe has not been appreciated. . 

Table op Lengths of Pipe in Feet That Give Resistance Equal That 
op Various Fittings 



Diameter 
of Pipe 


90° Elbows 


Un reamed 

Joints, Two 

Ends 


Reamed Joints, 
Two Ends 


Return 
Bends 


Globe 
Valves 


1 

2 
3 
4 
1 

11 

2 


10.0 
7.0 
5.0 
4.0 
3.5 


2 to 4 
it 

it 

tt 

tt 


1.0 
1.0 
1.0 
1.0 
1.0 


10.0 
7.0 
5.0 
4.0 
3.5 


20.0 
25.0 
40.0 
45.0 
47.0 



A series of runs was made on 50-foot lengths of rubber-lined 
armored hose such as is used to connect with compressed-air tools. 
The scheme was the same as that described for pipes and fittings ; 

v ^ 
and the range of — was the same. The average results are here 

given. This includes the resistance in a 50-foot length with the 
metallic end couplings. In these end connections a considerable 
contraction occurs. For the half-inch hose the end couplings are 
quarter-inch. The excessive resistance in the half-inch hose may 
have been due to these end contractions or to some other ob- 
struction. It is a further illustration of the fact that reliable 
coefficients cannot be gotten for pipes of half-inch diameter and 
less. 



Diameter of hose in inches 
Resistance in 50-ft. lengths 


i 

2 

950.0^ 
r 


3 

4 

20.0 — 
r 


1 

4.5^ 
r 


14 

2.6^ 
r 



APPENDIX D 
THE OIL DIFFERENTIAL GAGE 

Examination of Eq. (21) shows that the greatest liability to 
inaccuracy lies in determining i since it is relatively small as com- 
pared with t and p and the conditions are such that the scale can- 
not be read with precision. To better determine i the oil differ- 
ential gage may be used. A special design suitable to this 
purpose is illustrated in Fig. 32. l 

The special fittings are inserted in place of the two plain glass 
tubes of Fig. 32. The cocks being lettered similarly in the two 
figures. 

The special features of the gage are the two reservoirs G\ and 
Gi the capacity of each being controlled by the movable piston. 

The manipulation is as follows : With water in the low pressure 
side and oil in the high pressure side and with C 2 and Cz open, the 

specific gravity of the oil is -~- = s. Now with C 2 and C 3 closed 

and d and C5 open the higher pressure will depress the surface, 
Ai, of the oil and raise A 2, that of the water. Now, by manipu- 
lating the piston G\ oil can be forced in or withdrawn from the 
gage tube until A v and A\ are in the same horizontal plane, or on 
the same scale line. While this coincidence holds i = Z (1 — s). 
Proof.— Let w equal pressure due to 1 in. of water head and 
equal that due to 1 in. of oil head. Then since the two 
pressures become equal on the line BB, we have 

p -f- OZ = p% + wZo and pi — p 2 = Z Q (w — 0) 

but i = Vl ~ V2 - Therefore, i = Z (1 - s). 
w .- '. . 

With kerosene oil in the gage i equals one-fifth of Z very nearly. 

The length of oil and water in the gage tubes can be further con- 
trolled by the drain cocks, on the reservoirs. The length of oil 
should be about five times as much as the anticipated i and this 
(i) must be kept within the limits specified by the standards of 

1 This gage was designed and is used in the laboratories of the Missouri 
School of Mines. 

187 



188 



APPENDIX D 

Cs S ? ( i } " C<i 




Fig. 32. 



APPENDIX D 189 

practice. Say within 12 in. Gage tubes about 4 ft. long will be 
found convenient. 

The small pipes e\ and e 2 connect with the air main and so 
practically balance the pressures on the pistons in the reservoirs. 
Thus their movements are made easier and leakage by the pistons 
practically eliminated. 



INDEX 



Adiabatic compression, 3 
Adjustment of valves, 15 
Air-lift pump — Chart, 81 

Data on, 85 

Designing, 78 

Dredge, 83 

Testing wells, 84 

Theory, 76 
Air used without expansion — Effi- 
ciency, 30 
Altitude — Variation of pressure with, 

31 
Atmospheric air — Weights, 131 

Pressure — Variation with alti- 
tude, 31 

B 

Blowers— Rotary, 113 
C 

Centrifugal air compressors, 105 

Chart— Air-lift pump, 81 

Design of logarithmic, 172 
Friction in air pipes, 53 
Friction coefficients, 52 
Performance of fans, 102 

Clearance — Effect in Compression, 13 
Expansion engines, 14 
From indicator cards, 11 

Coefficients for friction formulas, 52 
Orifices, 38, 139 

Compounding, 23 

Proportions for, 25 

Compounding — Work in compres- 
sion, 27, 136 

Cooling — Effect on efficiency, 21 

Cooling water required, 20 

Compressed air reservoirs, 89 



D 

Density of air, 19, 131 
Displacement pumps, 75 
Drill capacities, 169 

E 

Efficiency-Power transmission, 60 

Volumetric, 13, 17 

Working without expansion, 30, 
138 
Elevation and pressure table, 138 
Exhaust pumps, 29 



Fans, 91 

Designing, 98 
Performance chart, 102 
Suggestions on designing, 103 
Flow through orifices, 35-44 
Friction — In air pipes, 49-55 
Chart for, 53 
Experiments in, 179 
Table for, 141 
Gas pipes, 55 
In pipe fittings, 55 



H 



Heating — -Effect on Volumetric effi- 
ciency, 17 
Effect on work efficiency, 21 
Hydraulic air compressors — Dis- 
placement type, 63, 139 
Entanglement type, 64 



Indicator cards, 9 
Isothermal compression, 1 



191 



192 



INDEX 



Logarithms — Hyperbolic, 147 

M 

Measurement of air, 33 

By standard orifice, 35 
In closed tanks, 46 
Under pressure, 41 

Moisture in air, 19 
Tables, 134 

O 

Oil differential gage, 187 
Orifice — Apparatus, 37 
Chart for sizes, 39 
Coefficients, 38, 43 
Tables of, 139, 140 



Pipes — Dimensions of, 146 

Planimeter constants, 9 

Power transmission by compressed 

air, 60 
Pump— Air-lift, 76 
Pump — Design of a mine pump, 115 

Design of displacement pump, 
120 

Displacement, 75 

Exhaust, 29 



R 



Reheating, 21 



Return-air pump, 69 
Return-air system, 68 

S 

Standard orifice, 35 

T 

Tables — See Subjects. 
Temperature changes, 18 
Temperature Table, 129 
Turbo air compressors — Formulas, 
107 
Proportions, 111 



Valve — Adjustments, 15 
Variable intake pressure, 27 
Volumetric efficiency, 13, 17 
Volume — Pressure, Temperature, 
Work, Table, 129 

W 

Water-cooling, 20 

Water gages, 40 

Weight of air, 19, 131 

Work — Adiabatic, 3 

In terms of temperature, 4 
Incomplete expansion, 7 
Isothermal conditions, 1 
Variable intake pressure, 27 



Chart for Solving Formula /=- - 10 25f, « 2 , or v* = 35.13 (fr) d *' 



rd 6 ' 31 i3600* 



/= Friction Loss in Pounds per Square Inch, 

I = Length of Pipe in Feet. 

v= Cubic Feet of Free Air per Minute. 

f- Ratio of Compression 

d- Diameter of Pipe in Inches. 

The Dependent Factors (fr) , V and d Lie in 
a Straight Line. To get the Friction Loss in 
1000 Feet; Divide the (fr) by r. 

Friction of Gasses will be Proportional 
the their Specific Gravities. 



80,000 
75,000- 
70,000- 
65,000 
60,000- 
55,000 
50,000 
45,000- 
40,000- 
35,000- 
30,000- 
28,000 
26,000- 
24,000- 
22 000 - 
20,000- 
18,000 
16,000- 
14,000- 
12.000- 
10,000 
9000 
8000 
7000- 
6000- 
5000- 
4500- 
4000 
3500- 
3000- 

2500- 

2000. 
1800- 
1600- 
1400 
1200- 

1000. 
900 
800- 
700- 
600 

500. 
450 
400 
350 
300 



00- 


u 


90- 


o 

a 


80- 


u 






70- 


< 


60- 


o 
2 




^ 



Plate III. 



10 



8- 



6- 



4ft- 



4 - 



24- 



2- 



1H- 



14- 



1M- 



i- 



